The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁
We define certain extensions of Jacobi groups of A₁, prove an analogue of the Chevalley theorem for their invariants, and construct a Dubrovin-Frobenius structure on their orbit space.
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2021 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211166 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁. Guilherme F. Almeida. SIGMA 17 (2021), 022, 39 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859753770815062016 |
|---|---|
| author | Almeida, Guilherme F. |
| author_facet | Almeida, Guilherme F. |
| citation_txt | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁. Guilherme F. Almeida. SIGMA 17 (2021), 022, 39 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We define certain extensions of Jacobi groups of A₁, prove an analogue of the Chevalley theorem for their invariants, and construct a Dubrovin-Frobenius structure on their orbit space.
|
| first_indexed | 2026-03-15T18:30:06Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 022, 39 pages
The Differential Geometry of the Orbit Space
of Extended Affine Jacobi Group A1
Guilherme F. ALMEIDA
SISSA, via Bonomea 265, Trieste, Italy
E-mail: galmeida@sissa.it
Received May 30, 2020, in final form February 11, 2021; Published online March 09, 2021
https://doi.org/10.3842/SIGMA.2021.022
Abstract. We define certain extensions of Jacobi groups of A1, prove an analogue of
Chevalley theorem for their invariants, and construct a Dubrovin–Frobenius structure on its
orbit space.
Key words: Dubrovin–Frobenius manifolds; Hurwitz spaces; extended Jacobi groups
2020 Mathematics Subject Classification: 53D45
Dedicated to the memory of Professor Boris Dubrovin
1 Introduction
Dubrovin–Frobenius manifold is a geometric interpretation of a remarkable system of differential
equations called WDVV equations [6]. Since early nineties, there has been a continuous exchange
of ideas from fields that are not trivially related to each other, such as topological quantum field
theory, non-linear waves, singularity theory, random matrices theory, integrable systems, and
Painlevé equations. Dubrovin–Frobenius manifolds theory is a bridge between them.
1.1 Orbit space of reflection groups and its extensions
In [6], Dubrovin pointed out that WDVV solutions with certain good analytic properties are rela-
ted with partition functions of TFT. Afterwards, Dubrovin conjectured that WDVV solutions
with certain good analytic properties are in one to one correspondence with discrete groups. This
conjecture is supported by ideas which come from singularity theory, because in this setting
there exists an integrable systems/discrete group correspondence. Furthermore, in minimal
models, such as Gepner chiral rings, there exists a correspondence between physical models
and discrete groups. In [14], Hertling proved that a particular class of Dubrovin–Frobenius
manifold, called polynomial Dubrovin–Frobenius manifold, is isomorphic to the orbit space of
a finite Coxeter group, which are spaces such that their geometric structure is invariant under the
finite Coxeter group. In [2, 3, 6, 7, 9, 10, 24], there are many examples of WDVV solutions that
are associated with orbit spaces of natural extensions of finite Coxeter groups, such as extended
affine Weyl groups, and Jacobi groups. Therefore, the construction of Dubrovin–Frobenius
manifolds on orbit space of reflection groups and its extensions is a prospective project of the
classification of WDVV solutions. In addition, WDDV solutions arising from orbit spaces may
also have some applications in TFT or some combinatorial problem, because previously these
relationships were demonstrated in some examples, such as the orbit space of the finite Coxeter
group A1, and the extended affine Weyl group A1 [8, 11].
mailto:galmeida@sissa.it
https://doi.org/10.3842/SIGMA.2021.022
2 G.F. Almeida
1.2 Hurwtiz space/orbit space correspondence
There are several other non-trivial connections that Dubrovin–Frobenius manifolds theory can
make. For example, Hurwitz spaces is the one of the main sources of examples of Dubrovin–
Frobenius manifolds. Hurwitz spaces Hg,n0,n1,...,nm are moduli space of covering over CP1 with
a fixed ramification profile. More specifically, Hg,n0,n1,...,nm is moduli space of pairs{
Cg, λ : Cg 7→ CP1
}
,
where Cg is a compact Riemann surface of genus g and λ is meromorphic function with poles in
λ−1(∞) = {∞0,∞1, . . . ,∞m}.
Moreover, λ has degree ni + 1 near ∞i. Hurwitz space, with a choice of a specific Abelian
differential, called quasi-momentum or primary differential, give rise to a Dubrovin–Frobenius
manifold; see section [6, 19] for details. In some examples, the Dubrovin–Frobenius structure
of Hurwitz spaces are isomorphic to Dubrovin–Frobenius manifolds associated with orbit spaces
of suitable groups. For instance, the orbit space of the finite Coxeter group A1 is isomorphic
to the Hurwitz space H0,1. Furthermore, orbit space of the extended affine Weyl group Ã1 and
of the Jacobi group J (A1) are isomorphic to the Hurwitz spaces H0,0,0 and H1,1 respectively.
Motivated by these examples, we construct the following diagram
H0,1
∼= orbit space of A1 H0,0,0
∼= orbit space of Ã1
H1,1
∼= orbit space of J (A1) H1,0,0
∼= ?
1
2 4
3
From the Hurwitz space side, the vertical lines 2 and 4 mean that we increase the genus by 1,
and the horizontal lines mean that we split one pole of order 2 into two simple poles. From the
orbit space side, the vertical line 2 means that we are doing an extension from the finite Coxeter
group A1 to the Jacobi group J (A1); the line horizontal line 1 means that we are extending the
orbit space of A1 to the extended affine Weyl group Ã1. Therefore, one might ask if the line 3
and 4 would imply an orbit space interpretation of the Hurwitz space H1,0,0. The main goal
of this paper is to define a new class of groups such that its orbit space carries the Dubrovin–
Frobenius structure of H1,0,0. The new group is called extended affine Jacobi group An, and is
denoted by J
(
Ã1
)
. This group is an extension of the Jacobi group J (A1) and of the extended
affine Weyl group Ã1.
1.3 Results
The main goal of this paper is to construct the Dubrovin–Frobenius structure of the Hurwitz
space H1,0,0 from the data of the group J
(
Ã1
)
. In other words, we derive the WDVV solution
associated to the group J
(
Ã1
)
without using the correspondent Hurwitz space construction.
First of all, recall the definition of WDVV equation:
Definition 1.1. The function F (t), t =
(
t1, t2, . . . , tn
)
is a solution of a WDVV equation if its
third derivatives
cαβγ =
∂3F
∂tα∂tβ∂tγ
(1.1)
satisfy the following conditions:
1) ηαβ = c1αβ is constant nondegenerate matrix;
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 3
2) the function
cγαβ = ηγδcαβδ
is structure constant of associative algebra;
3) F (t) must be quasi-homogeneous function
F
(
cd1t1, . . . , cdntn
)
= cdFF
(
t1, . . . , tn
)
for any nonzero c and for some numbers d1, . . . , dn, dF .
Our goal is to extract a WDVV equation from the data of a suitable group J
(
Ã1
)
. We define
the group J
(
Ã1
)
. Recall that the group A1 acts on C 3 v0 by reflections
v0 7→ −v0.
The group J
(
Ã1
)
is an extension of the group A1 in the following sense:
Proposition 1.2. The group J
(
Ã1
)
3 (w, t, γ) acts on Ω := C⊕C2⊕H 3 (u, v, τ) = (u, v0, v2, τ)
as follows:
w(u, v, τ) = (u,wv, τ),
t(u, v, τ) =
(
u− 〈λ, v〉Ã1
− 1
2
〈λ, λ〉Ã1
τ, v + λτ + µ, τ
)
,
γ(u, v, τ) =
(
u+
c〈v, v〉Ã1
2(cτ + d)
,
v
cτ + d
,
aτ + b
cτ + d
)
,
where w ∈ A1 acts by reflection in the first v0 variables of C2 3 v = (v0, v2),
t = (λ, µ) ∈ Z2,
(
a b
c d
)
∈ SL2(Z), 〈v, v〉Ã1
= 2v20 − 2v22.
See Section 2.1 for details.
In order to define any geometric structure in an orbit space, first it is necessary to define
a notion of invariant J
(
Ã1
)
sections. For this purpose, we generalise the ring of invariant
functions used in [2, 3] for the group J (A1), which are called Jacobi forms. This notion was
first defined in [12] by Eichler and Zagier for the group J (A1), and it was further generalised
for the group J (A1) in [23] by Wirthmuller. Furthermore, an explicit base of generators were
derived in [2, 3] by Bertola. The Jacobi forms used in this thesis are defined by:
Definition 1.3. The weak Ã1-invariant Jacobi forms of weight k, order l, and index m are
functions on Ω = C⊕ Cn+2 ⊕H 3 (u, v0, v2, τ) = (u, v, τ) which satisfy
ϕ(w(u, v, τ)) = ϕ(u, v, τ), A1-invariant condition,
ϕ(t(u, v, τ)) = ϕ(u, v, τ),
ϕ(γ(u, v, τ)) = (cτ + d)−kϕ(u, v, τ),
Eϕ(u, v, τ) := − 1
2πi
∂
∂u
ϕ(u, v, τ) = mϕ(u, v, τ), Euler vector field. (1.2)
Moreover, the weak Ã1-invariant Jacobi forms are meromorphic in the variable v2 on a fixed
divisor, in contrast with the Jacobi forms of the group J (A1) ,which are holomorphic in each
variable; see details on Section 2.2. The ring of weak Ã1-invariant Jacobi forms gives the notion
4 G.F. Almeida
of the Euler vector field; indeed, the vector field defined in the last equation of (1.2) measures
the degree of the Jacobi forms, which coincides with the index. The differential geometry of
the orbit space of the group J
(
Ã1
)
should be understood as the space such that its sections
are written in terms of Jacobi forms. Then, in order for this statement to make sense, we must
prove a Chevalley type theorem, which is:
Theorem 1.4. The trigraded algebra of Jacobi forms J
J (Ã1)
•,•,• =
⊕
k,l,m J
Ã1
k,l,m is freely generated
by 2 fundamental Jacobi forms (ϕ0, ϕ1) over the graded ring E•,•
J
J (Ã1)
•,•,• = E•,• [ϕ0, ϕ1] ,
where
E•,• = J•,•,0
is the ring of coefficients.
More specifically, the ring of function E•,• is the space of functions f(v2, τ) such that, for
fixed τ , the functions τ 7→ f(v2, τ) is an elliptic function.
Moreover, (ϕ0, ϕ1) are given by
Corollary 1.5. The function[
e
z ∂
∂p
(
e2πiu
θ1(v0 + v2 + p)θ1(−v0 + v2 + p)
θ1(2v2 + p)θ′1(0)
)] ∣∣∣∣
p=0
= ϕ
J (Ã1)
1 + ϕ
J (Ã1)
0 z +O
(
z2
)
,
generates the Jacobi forms ϕ
J (Ã1)
0 and ϕ
J (A1)
1 , where
ϕ
J (Ã1)
0 :=
∂
∂p
(
ϕ̂
J (Ã1)
1
)∣∣∣∣
p=0
.
This lemma realises the functions (ϕ0, ϕ1, v2, τ) as coordinates of the orbit space of J
(
Ã1
)
.
The unit vector field is chosen to be
e =
∂
∂ϕ0
, (1.3)
because ϕ0 is the basic generator with maximum weight degree; see the Section 2.2 for details.
The last component we need to construct is the intersection form of the orbit space of J
(
Ã1
)
.
The natural candidate to be such a metric is the invariant metric of the group J
(
Ã1
)
, which
given by
g = 2dv20 − 2dv22 + 2dudτ. (1.4)
From the data of the intersection form (1.4), is possible to derive a second flat metric of the
orbit space J
(
Ã1
)
. The second metric is given by
η∗ := Lieeg
∗,
and it is denoted by the Saito metric due to K. Saito, who was the first to define this metric
for the case of finite Coxeter group [18]. One of the main technical problems of this paper is to
prove that the Saito metric η∗ is flat. At this point, we can state our main result.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 5
Theorem 1.6. A suitable covering of the orbit space
(
C⊕ C2 ⊕H
)
/J
(
Ã1
)
with the intersection
form (1.4), unit vector field (1.3), and Euler vector field given by the last equation of (1.2) has
a Dubrovin–Frobenius manifold structure. Moreover, a suitable covering of C⊕Cn+1⊕H/J
(
Ã1
)
is isomorphic as Dubrovin–Frobenius manifold to a suitable covering of the Hurwitz space H1,0,0.
See Section 3.4 for details. In particular, we derive explicitly the WDVV solution associated
with the orbit space of J
(
Ã1
)
, which is given by
F
(
t1, t2, t3, t4
)
=
i
4π
(
t1
)2
t4 − 2t1t2t3 −
(
t2
)2
log
(
t2
θ′1
(
0, t4
)
θ1
(
2t3, t4
)) ,
where
θ1(v, τ) = 2
∞∑
n=0
(−1)neπiτ(n+
1
2
)2 sin((2n+ 1)v). (1.5)
The results of this paper are important because of the following:
1. The Hurwitz spaces H1,0,0 are classified by the group J
(
Ã1
)
, hence we increase the know-
ledge of the WDVV/discrete group correspondence. Recently, the case J
(
Ã1
)
attracted
the attention of experts, due to its application in integrable systems [5, 13, 16].
2. The orbit space construction of the group J
(
Ã1
)
can be generalised to the group J (Ãn);
see the definition in [1]. Further, the same can be done to the other classical finite Coxeter
groups as Bn, Dn. Hence, these orbit spaces could give rise to a new class of Dubrovin–
Frobenius manifolds. Furthermore, the associated integrable hierarchies of this new class
of Dubrovin–Frobenius manifolds could have applications in Gromow–Witten theory and
combinatorics.
This paper is organised in the following way: In Section 2, we define extended affine Jacobi
group J
(
Ã1
)
and we prove some results related with its ring of invariant functions. In Sec-
tion 3, we construct a Dubrovin–Frobenius structure on the orbit spaces of J
(
Ã1
)
and compute
its free-energy. Furthermore, we show that the orbit space of the group J
(
Ã1
)
is isomorphic,
as a Dubrovin–Frobenius manifold, to the Hurwitz–Frobenius manifold H̃1,0,0 [6, 19]. See The-
orem 3.7 for details.
2 Invariant theory of J
(
Ã1
)
The focus of this section is to define a new extension of the finite Coxeter group A1 such
that it contains the affine Weyl group Ã1 and the Jacobi group J (A1). This new extension
will be denoted by Extended affine Jacobi group J
(
Ã1
)
. Further, we prove that, from the
data of the group J
(
Ã1
)
, we can reconstruct the Dubrovin–Frobenius structute of the Hurwitz
space H1,0,0 on the orbit space of J
(
Ã1
)
. The advantage of this orbit space construction
is the Chevalley Theorem 2.25, which gives a global interpretation for orbit space of J
(
Ã1
)
.
Furthermore, it attaches the group J
(
Ã1
)
to the Hurwitz space H1,0,0, and this fact might
be useful in the general understanding of WDVV/group correspondence. These results sre
interesting because the Hurwitz space H1,0,0 is well know to have a rich Dubrovin–Frobenius
structure, called a tri-Hamiltonian structure [16] and [15]. This fact realises the orbit space of
J
(
Ã1
)
as suitable ambient space for Dubrovin–Frobenius submanifolds. Furthermore, it shows
an interesting relationship relation between the integrable systems of the ambient space and the
integrable systems of its Dubrovin–Frobenius submanifolds.
6 G.F. Almeida
2.1 The group J
(
Ã1
)
The main goal of this section is to motivate and to define the group J
(
Ã1
)
. In order to do
that, it will be necessary to recall the definition of the group A1, and some of its extensions.
Moreover, its goal is to understand how to derive WDDV solution starting from these groups.
The group An acts on the space ΩAn =
{
(v0, v1, . . . , vn) ∈ Cn+1 :
∑n
i=0 vi = 0
}
by permuta-
tions:
(v0, v1, . . . , vn) 7→ (vi0 , vi1 , . . . , vin). (2.1)
Let us concentrate on the simplest possible case, i.e., n = 1. In this case, the action on C ∼= ΩA1
is just:
v0 7→ −v0.
The understanding of the orbit space of A1 requires a Chevalley theorem for the ring of invariants.
The Chevalley theorem form the group An says that
Theorem 2.1 ([4]). Let the Coxeter group An which acts on ΩAn 3 (v0, v1, . . . , vn) as (2.1),
then
C[v0, v1, . . . , vn]An ∼= C[a2, a3, . . . , an+1],
where ai are weighted homogeneous polynomials of degree i.
In the A1 case, the ring of invariants is just
C
[
v20
] ∼= C[a2],
then the orbit space of A1 is just the
Spec
(
C
[
v20
])
.
In the papers [6, 7], it was demonstrated that C/A1 has structure of Dubrovin–Frobenius mani-
fold. Furthermore, it is isomorphic to the Hurwitz space H0,1, i.e., the space of rational functions
with a double pole. The isomorphism can be realized by the following map:
[v0] 7→ λA1(p, v0) = (p− v0)(p+ v0) = p2 + a2.
Note that the isomorphism works, because λA1(p, v0) is invariant under the A1-action. Applying
the methods developed in [6, 7], one can show that the WDVV solution associated with this
orbit space is
F
(
t1
)
=
(
t1
)3
6
,
where t1 is the flat coordinate of the metric η.
In [6, 10] it was also considered the extended affine A1 that is denoted by Ã1. The action on
(
LA1 ⊗ C
)
⊕ C =
{
(v0, v1, v2) ∈ C3 :
1∑
i=0
vi = 0
}
is
v0 7→ ±v0 + µ0, v2 7→ v2 + µ2,
where µ0, µ2 ∈ Z.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 7
A notion of the invariant ring for the group extended affine An was defined in [10], and
Dubrovin and Zhang proved that this invariant ring for the case Ã1 is isomorphic to
C
[
e2πiv2 cos(2πiv0), e
2πiv2
]
.
Therefore, the orbit space of Ã1 is the weight projective variety associated with
Spec
(
C
[
e2πiv2 cos(2πiv0), e
2πiv2
])
.
Further, a Dubrovin–Frobenius manifold structure was built on the orbit space of Ã1 with the
following WDVV solution:
F
(
t1, t2
)
=
(
t1
)2
t2
2
+ et
2
. (2.2)
The orbit space of Ã1 is also associated with a Hurwitz space, but the relation is slightly less
straightforward. The first step is to consider the following map:
[v0, v2] 7→ λÃ1(p, v0, v2) = ep + e2πiv2 cos(2πiv0) + e2πiv2e−p.
The second is to consider the Legendre transformation of S2 type [6, Appendix B and Chapter 5].
Consider
b = e2πiv2 cos(2πiv0), a = e2πiv2 ,
and the following choice of primary differential dp̃ implicity given by
dp =
dp̃
p̃− b
.
Then, in these new coordinates λÃ1 , is given by
λ(p̃, a, b) = p̃+
a
p̃− b
.
Hence, the orbit space of Ã1 is isomorphic to the Hurwitz space H0,0,0, i.e., space of fractional
functions with two simple poles.
The next example of group to be considered is the Jacobi group J (A1), which acts on
ΩJ (A1) :=
(
LA1 ⊗ C
)
⊕ C⊕H =
{
(v0, v1, u, τ) ∈ C3 ⊕H :
1∑
i=0
vi ∈ Z + τZ
}
as follows:
A1-action:
v0 7→ −v0, u 7→ u, τ 7→ τ. (2.3)
Translation:
v0 7→ v0 + µ0 + λ0τ, u 7→ u− λ0v0 −
λ20
2
τ, τ 7→ τ, (2.4)
where µ0, λ0 ∈ Z.
SL2(Z)-action:
v0 7→
v0
cτ + d
, u 7→ u− cv20
2(cτ + d)
, τ 7→ aτ + b
cτ + d
, (2.5)
where a, b, c, d ∈ Z, and ad− bc = 1.
The notion of invariant ring of J (A1) was first defined in [12]. However, the definitions stated
in [2, 3, 23] are more suitable for this purpose.
8 G.F. Almeida
Definition 2.2. The weak A1-invariant, Jacobi forms of weight k, and index m are holomorphic
functions on Ω = C⊕ C⊕H 3 (u, v0, τ) which satisfy
ϕ(u,−v0, τ) = ϕ (u, v0, τ) , A1-invariant condition,
ϕ
(
u− λ0v0 −
λ0
2
2
τ, v0 + λ0τ + µ, τ
)
= ϕ (u, v0, τ) ,
ϕ
(
u+
cv20
2(cτ + d)
,
v0
cτ + d
,
aτ + b
cτ + d
)
= (cτ + d)k ϕ(u, v0, τ),
Eϕ(u, v0, τ) :=
1
2πi
∂
∂u
ϕ(u, v0, τ) = mϕ(u, v0, τ).
Moreover, ϕ are locally bounded functions of v0 as =(τ) 7→ +∞ (weak condition).
The space of Ã1-invariant Jacobi forms of weight k, and index m is denoted by JA1
k,m, and
J
J (A1)
•,• =
⊕
k,m J
A1
k,m is the space of Jacobi forms A1-invariant.
In [12], it was proved the following a version of the Chevalley theorem.
Theorem 2.3. Let J
J (A1)
•,• the ring of Jacobi forms A1-invariant, then
J
J (A1)
•,• ∼= M•[ϕ0, ϕ2],
where M• is the ring of holomorphic modular forms, and
ϕ2 = e2πiu
(
θ1(v0, τ)
θ′1(0, τ)
)2
, ϕ0 = ϕ2℘(v0, τ),
θ1 is the Jacobi θ1-function (1.5), and ℘ is the Weierstrass P-function, which is defined as
℘(v, τ) =
1
v2
+
∞∑
m2+n2 6=0
1
(v −m− nτ)2
− 1
(m+ nτ)2
. (2.6)
Note that this Chevalley theorem is slightly different from the others. The ring of the coef-
ficients is the ring of holomorphic of modular forms, instead of just C. The geometric interpre-
tation of this fact is that the orbit space of J (A1) is a line bundle, such that its base is family
of elliptic curves Eτ quotient by the group A1 parametrised by H/SL2(Z). In [2] and [3], it was
proved that orbit space of J (A1) has a Dubrovin–Frobenius structure. Furthermore, the orbit
space of J (A1) is isomorphic to H1,1, i.e., space of elliptic functions with one double pole. The
explicit isomorphism is given by the map
[(u, v0, τ)] 7→ λJ (A1)(v, u, v0, τ) = e2πiu
θ1(v − v0, τ)θ1(v + v0, τ)
θ21(v, τ)
. (2.7)
As in the A1 case, the isomorphism is only possible, because the map (2.7) is invariant un-
der (2.3)–(2.5). A WDVV solution for this case is the following:
F
(
t1, t2, τ
)
=
(
t1
)2
τ
2
+
t1
(
t2
)2
2
−
πi
(
t2
)2
48
E2(τ), (2.8)
where
E2(τ) = 1 +
3
π2
∑
m6=0
∞∑
n=−∞
1
(m+ nτ)2
.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 9
A remarkable fact in these orbit space constructions is its correspondence with Hurwitz spaces,
which can be summarized by the following diagram:
H0,1
∼= C/A1 H0,0,0
∼= C2/Ã1
H1,1
∼= (C⊕ C⊕H) /J (A1) H1,0,0
∼= ?
1
2 4
3
The arrows of the diagram above have a double meaning. The first one is simply an extension
of the group, the arrow 2 is “Jacobi” extension, and the arrow 1 is “affine” extension. The second
meaning is related to the Hurwitz space side: the arrows 2 and 4 increase by one the genus, and
the arrows 1 and 3 split a double pole in 2 simple poles. The missing part of the diagram is
exactly the orbit space counter part of H1,0,0. The diagram suggest that the new group should
be an extension of the A1 group, such that combine the groups Ã1, and J (A1), furthermore,
it should preserve H1,0,0 in a similar way for what was done in (2.7). To construct the desired
group, we start from the group J (A1) and make an extension in order to incorporate the Ã1
group. Concretely, we extend the domain ΩJ (A1) to
ΩJ (Ã1) := ΩA1 ⊕ C⊕ C⊕H =
{
(v0, v1, v2, u, τ) ∈ C4 ⊕H : v0 + v1 ∈ Z⊕ τZ
}
,
and we extend the group action J (A1) to the following action:
A1-action:
v0 7→ −v0, v2 7→ v2, u 7→ u, τ 7→ τ. (2.9)
Translation:
v0 7→ v0 + µ0 + λ0τ, v2 7→ v2 + µ2 + λ2τ,
u 7→ u− 2λ0v0 + 2λ2v2 − λ20τ + λ22τ + k, τ 7→ τ, (2.10)
where (λ0, λ2), (µ0, µ2) ∈ Z2, and k ∈ Z.
SL2(Z)-action:
v0 7→
v0
cτ + d
, v2 7→
v2
cτ + d
, u 7→ u+
c
(
v20 − v22
)
(cτ + d)
, τ 7→ aτ + b
cτ + d
, (2.11)
where a, b, c, d ∈ Z, and ad− bc = 1.
The group action (2.9), (2.10), and (2.11) is called extended affine Jacobi group A1, and is
denoted by J
(
Ã1
)
.
Remark 2.4. The translations of the group Ã1 are a subgroup of the translations of the
group J
(
Ã1
)
. Therefore, it is in that sense that J
(
Ã1
)
is a combination of Ã1 and J (A1).
In order to rewrite the action of J
(
Ã1
)
in an intrinsic way, consider the A1 in the following
extended space
LÃ1 =
{
(z0, z1, z2) ∈ Z3 :
3∑
i=0
zi = 0
}
.
The action of A1 on LÃ1 is given by
w(z0, z1, z2) = (z1, z0, z2)
10 G.F. Almeida
permutations in the first two variables. Moreover, A1 also acts on the complexfication of LÃ1⊗C.
Let us use the following identification Z2 ∼= LÃ1 , C2 ∼= LÃ1 ⊗ C, which is possible due to the
maps
(v0, v2) 7→ (v0,−v0, v2), (v0, v1, v2) 7→ (v0, v2).
The action of A1 on C2 3 v = (v0, v2) is:
w(v) = w(v0, v2) = (−v0, v2).
Let the quadratic form 〈 , 〉Ã1
be given by
〈v, v〉Ã1
= vTMÃ1
v = vT
(
2 0
0 −2
)
v = 2v20 − 2v22. (2.12)
Consider the following group LÃ1 × LÃ1 × Z with the following group operation
∀(λ, µ, k),
(
λ̃, µ̃, k̃
)
∈ LÃ1 × LÃ1 × Z,
(λ, µ, k) •
(
λ̃, µ̃, k̃
)
=
(
λ+ λ̃, µ+ µ̃, k + k̃ + 〈λ, λ̃〉Ã1
)
.
Note that 〈, 〉Ã1
is invariant under A1 group, then A1 acts on LÃ1 × LÃ1 × Z. Hence, we can
take the semidirect product A1 n
(
LÃ1 × LÃ1 × Z
)
given by the following product
∀(w, λ, µ, k),
(
w̃, λ̃, µ̃, k̃
)
∈ A1 × LÃ1 × LÃ1 × Z,
(w, λ, µ, k) •
(
w̃, λ̃, µ̃, k̃
)
=
(
ww̃,wλ+ λ̃, wµ+ µ̃, k + k̃ + 〈λ, λ̃〉Ã1
)
.
Denoting W (Ã1) := A1 n
(
LÃ1 × LÃ1 × Z
)
, we can define
Definition 2.5. The Jacobi group J
(
Ã1
)
is defined as a semidirect product W
(
Ã1
)
o SL2(Z).
The group action of SL2(Z) on W (Ã1) is defined as
Adγ(w) = w,
Adγ(λ, µ, k) =
(
aµ− bλ,−cµ+ dλ, k +
ac
2
〈µ, µ〉Ã1
− bc〈µ, λ〉Ã1
+
bd
2
〈λ, λ〉Ã1
)
for (w, t = (λ, µ, k)) ∈W
(
Ã1
)
, γ ∈ SL2(Z). Then the multiplication rule is given as follows
(w, t, γ) •
(
w̃, t̃, γ̃
)
=
(
ww̃, tAdγ(wt̃), γγ̃
)
.
Then, the action of Jacobi group J
(
Ã1
)
on ΩJ (Ã1) := C⊕C2 ⊕H ∈ (u, v, τ) is described by
the main three generators
ŵ =
(
w, 0, ISL2(Z)
)
, t =
(
IA1 , λ, µ, k, ISL2(Z)
)
, γ =
(
IA1 , 0,
(
a b
c d
))
,
which acts on ΩJ (Ã1) as follows
ŵ(u, v = (v0, v2), τ) = (u,−v0, v2, τ),
t(u, v = (v0, v2), τ) =
(
u− 〈λ, v〉Ã1
− 1
2
〈λ, λ〉Ã1
τ + k, v0 + λ0τ + µ0, v2 + λ2τ + µ2, τ
)
,
γ(u, v = (v0, v2), τ) =
(
u+
c〈v, v〉Ã1
2(cτ + d)
,
v0
cτ + d
,
v2
cτ + d
,
aτ + b
cτ + d
)
,
where λ, µ, k ∈ LÃ1 × LÃ1 × Z,
λ = (λ0, λ2), µ = (µ0, µ2).
In a more condensed form we have the following proposition.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 11
Proposition 2.6. The group J
(
Ã1
)
3 (ŵ, t, γ) acts on Ω := C⊕ C2 ⊕H 3 (u, v, τ) as follows:
ŵ(u, v, τ) = (u,w(v), τ),
t(u, v, τ) =
(
u− 〈λ, v〉Ã1
− 1
2
〈λ, λ〉Ã1
τ + k, v + λτ + µ, τ
)
,
γ(u, v, τ) =
(
u+
c〈v, v〉Ã1
2(cτ + d)
,
v
cτ + d
,
aτ + b
cτ + d
)
. (2.13)
Substituting (2.12) in (2.13), we get the transformation law (2.9), (2.10), and (2.11). The
explanation of why (2.13) is that a group action for J
(
Ã1
)
is just straightforward computations,
but it is a bit long, so this part of the proof will be omitted.
2.2 Jacobi forms of J
(
Ã1
)
In order to understand the differential geometry of orbit space, first we need to study the algebra
of the invariant functions. Informally, every time that there is a group W acting on a vector
space V , one could think of the orbit spaces V/W as V , but you should remember yourself one
can only use the W -invariant sections of V. Hence, motivated by the definition of Jacobi forms
of group An defined in [23], and used in the context of Dubrovin–Frobenius manifold in [2, 3],
we present the following:
Definition 2.7. The weak Ã1-invariant Jacobi forms of weight k, order l, and index m are
functions on Ω = C⊕ C2 ⊕H 3 (u, v0, v2, τ) = (u, v, τ) which satisfy
ϕ(w(u, v, τ)) = ϕ(u, v, τ), A1-invariant condition,
ϕ(t(u, v, τ)) = ϕ(u, v, τ),
ϕ(γ(u, v, τ)) = (cτ + d)−kϕ(u, v, τ),
Eϕ(u, v, τ) := − 1
2πi
∂
∂u
ϕ(u, v0, v2, τ) = mϕ(u, v0, v2, τ). (2.14)
Moreover,
1) ϕ is locally bounded functions of v0 as =(τ) 7→ +∞ (weak condition),
2) for fixed u, v0, τ the function v2 7→ ϕ(u, v0, v2, τ) is meromorphic with poles of order
at most l + 2m at in v2 = 0, 12 ,
τ
2 ,
1+τ
2 mod Z⊕ τZ,
3) for fixed u, v2 6= 0, 12 ,
τ
2 ,
1+τ
2 mod Z⊕τZ, τ the function v0 7→ ϕ(u, v0, v2, τ) is holomorphic,
4) for fixed u, v0, v2 6= 0, 12 ,
τ
2 ,
1+τ
2 mod Z ⊕ τZ. The function τ 7→ ϕ(u, v0, v2, τ) is holo-
morphic.
The space of Ã1-invariant Jacobi forms of weight k, order l, and index m are denoted by J Ã1
k,l,m,
and J
J (Ã1)
•,•,• =
⊕
k,l,m J
Ã1
k,l,m is the space of Jacobi forms Ã1-invariant.
Remark 2.8. The condition Eϕ(u, v0, v2, τ) = mϕ(u, v0, v2, τ) implies that ϕ(u, v0, v2, τ) has
the following form
ϕ(u, v0, v2, τ) = f(v0, v2, τ)e2πimu
and the function f(v0, v2, τ) has the following transformation law
f(v0, v2, τ) = f(−v0, v2, τ),
f(v0, v2, τ) = e−2πim
(
〈λ,v〉+ 〈λ,λ〉
2
τ
)
f(v0 +m0 + n0τ, v2 +m2 + n2τ, τ),
f(v0, v2, τ) = (cτ + d)−ke
2πim
(
c〈v,v〉
(cτ+d)
)
f
(
v0
cτ + d
,
v2
cτ + d
,
aτ + b
cτ + d
)
.
The functions f(v0, v2, τ) are more closely related to the definition of Jacobi form of the Eichler–
Zagier type [12]. The coordinate u works as kind of automorphic correction in this functions
12 G.F. Almeida
f(v0, v2, τ). Further, the coordinate u will be crucial to construct an equivariant metric on the
orbit space of J
(
Ã1
)
; see Section 3.
Remark 2.9. Note that the Jacobi forms in the Definition 2.2 are holomorphic, and in the
Definition 2.7, the Jacobi forms are meromorphic in the variable v2.
The main result of this section is the following:
The ring of Ã1-invariant Jacobi forms is polynomial over a suitable ring E•,• := J
J (Ã1)
•,•,0
on suitable generators ϕ0, ϕ1. Before stating precisely the theorem, I will define the objects
E•,•, ϕ0, ϕ1.
The ring E•,l := J
J (Ã1)
•,l,0 is the space of meromorphic Jacobi forms of index 0 with poles of
order at most l at 0, 1
2 , τ
2 , 1+τ
2 mod Z ⊕ τZ, by definition. The sub-ring J
J (Ã1)
•,0,0 ⊂ E•,• has
a nice structure, indeed:
Lemma 2.10. The sub-ring J
J (Ã1)
•,0,0 is equal to M• :=
⊕
Mk, where Mk is the space of modular
forms of weight k for the full group SL2(Z).
Proof. Using the Remark 2.8, we know that functions ϕ(u, v0, v2, τ) ∈ JJ (Ã1)
•,0,0 can not depend
on u, so ϕ(u, v0, v2, τ) = ϕ(v0, v2, τ). Moreover, for fixed v2, τ the functions v0 7→ ϕ(v0, v2, τ)
are holomorphic elliptic functions. Therefore, by Liouville theorem, these function are constant
in v0. Similar argument shows that these function do not depend on v2, because l + 2m = 0,
i.e., there is no pole. Then, ϕ = ϕ(τ) are standard holomorphic modular forms. �
Lemma 2.11. If ϕ ∈ E•,• = J
J (Ã1)
•,•,0 , then ϕ depends only on the variables v2, τ . Moreover,
if ϕ ∈ JJ (Ã1)
0,l,0 for fixed τ the function v2 7→ ϕ(v2, τ) is an elliptic function with poles of order
at most l on 0, 12 ,
τ
2 ,
1+τ
2 mod Z⊕ τZ.
Proof. The proof is essentially the same of the Lemma 2.10, the only difference is that now we
have poles at v2 = 0, 12 ,
τ
2 ,
1+τ
2 mod Z⊕ τZ. Hence, we have dependence on v2. �
As a consequence of Lemma 2.11, the function ϕ ∈ Ek,l = J
J (Ã1)
k,l,0 has the following form
ϕ(v2, τ) = f(τ)g(v2, τ),
where f(τ) is holomorphic modular form of weight k, and for fixed τ , the function v2 7→ g(v2, τ)
is an elliptic function of order at most l at the poles 0, 12 ,
τ
2 ,
1+τ
2 mod Z⊕ τZ.
At this stage, we are able to define ϕ0, ϕ1. Note that a natural way to produce meromorphic
Jacobi forms is by using rational functions of holomorphic Jacobi forms. Starting here, we will
denote the Jacobi forms related with the Jacobi group J (A1) with the upper index J (A1), for
instance
ϕJ (A1),
and the Jacobi forms related with the Jacobi group J
(
Ã1
)
with the upper index J
(
Ã1
)
ϕJ (Ã1).
In [2], Bertola found a basis of the generators of the Jacobi form algebra by producing
a holomorphic Jacobi form of type An as product of θ-functions
ϕJ (An) = e2πiu
n+1∏
i=1
θ1(zi, τ)
θ′1(0, τ)
.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 13
Afterwards, Bertola defined a recursive operator to produce the remaining basic generators.
In order to recall the details, see [2]. Our strategy will follow the same logic of Bertola method;
we use theta functions to produce a basic generator and thereafter, we produce a recursive
operator to produce the remaining part.
Lemma 2.12. Let be ϕ
J (A2)
3 (u1, z1, z2, τ) the holomorphic A2-invariant Jacobi form, which
corresponds to the algebra generator of maximal weight degree, in this case degree 3. More
explicitly,
ϕ
J (A2)
3 = e−2πiu1
θ1(z1, τ)θ1(z2, τ)θ1(−z1 − z2, τ)
θ′1(0, τ)3
.
Let be ϕ
J (A1)
2 (u2, z3, τ) the holomorphic A1-invariant Jacobi form, which corresponds to the
algebra generator of maximal weight degree, in this case degree 2:
ϕ
J (A1)
2 = e−2πiu2
θ1(z3, τ)2
θ′1(0, τ)2
.
Then, the function
ϕ
J (Ã1)
1 =
ϕ
J (A2)
3
ϕ
J (A1)
2
(2.15)
is meromorphic Jacobi form of index 1, weight −1, order 0.
Proof. For our convenience, we change the labels z1, z2, z3 to v0+v2, −v0+v2, 2v2, respectively.
Then (2.15) has the following form
ϕ
J (Ã1)
1 (u, v0, v2, τ) = e−2πiu
θ1(v0 + v2, τ)θ1(−v0 + v2, τ)
θ′1(0, τ)θ1(2v2, τ)
. (2.16)
Let us prove each item separately.
A1-invariant. The A1 group acts on (2.16) by permuting its roots, thus (2.16) remains
invariant under this operation.
Translation invariant. Recall that under the translation v 7→ v + m + nτ , the Jacobi theta
function transforms as [2, 22]:
θ1(vi + µi + λiτ, τ) = (−1)λi+µie−2πi
(
λivi+
λ2i
2
τ
)
θ1(vi, τ). (2.17)
Then, substituting the transformation (2.17) into (2.16), we conclude that (2.16) remains inva-
riant.
SL2(Z)-invariant. Under SL2(Z)-action the following function transform as
θ1
(
vi
cτ+d ,
aτ+d
cτ+d
)
θ′1
(
0, aτ+dcτ+d
) = (cτ + d)−1 exp
(
πicv2i
cτ + d
)
θ1(vi, τ)
θ′1(0, τ)
. (2.18)
Then, substituting (2.18) in (2.16), we get
ϕ
J (Ã1)
1 7→ ϕ
J (Ã1)
1
cτ + d
.
Index 1.
1
2πi
∂
∂u
ϕ1J
(
Ã1
)
= ϕ
J (Ã1)
1 .
Analytic behavior. Note that ϕ
J (Ã1)
1 θ21(2v2, τ) is holomorphic function in all the variables vi.
Therefore, ϕ
J (Ã1)
1 are holomorphic functions on the variables v0, and meromorphic function
in the variable v2 with poles on j
2 + lτ
2 , j, l = 0, 1 of order 2, i.e., l = 0, since m = 1. �
14 G.F. Almeida
In order to define the desired recursive operator, it is necessary to enlarge the domain of the
Jacobi forms from C ⊕ C2 ⊕ H 3 (u, v0, v2, τ) to C ⊕ C3 ⊕ H 3 (u, v0, v2, p, τ). In addition,
we define a lift of Jacobi forms defined in C⊕ C2 ⊕H to C⊕ C3 ⊕H as
ϕ(u, v0 + v2,−v0 + v2, τ) 7→ ϕ̂(p) := ϕ(u, v0 + v2 + p,−v0 + v2 + p, τ).
A convenient way to do computations in these extended Jacobi forms is to use the following
coordinates
s = u+ g1(τ)p2, z1 = v0 + v2 + p, z2 = −v0 + v2 + p, z3 = 2v2 + p, τ = τ.
The bilinear form 〈v, v〉Ã1
is extended to
〈(z1, z2, z3), (z1, z2, z3)〉E = z21 + z22 − z23 ,
or equivalently,
〈(v0, v2, p), (v0, v2, p)〉E = 2v20 − 2v22 + p2.
The action of the Jacobi group Ã1 in this extended space is
ŵE(u, v, p, τ) = (u,w(v), p, τ),
tE(u, v, p, τ) =
(
u− 〈λ, v〉E −
1
2
〈λ, λ〉Eτ + k, v + p+ λτ + µ, τ
)
,
γE(u, v, p, τ) =
(
u+
c〈v, v〉E
2(cτ + d)
,
v
cτ + d
,
p
cτ + d
,
aτ + b
cτ + d
)
.
Proposition 2.13. Let be ϕ ∈ JJ (Ã1)
k,m,• , and ϕ̂ the correspondent extended Jacobi form. Then,
∂
∂p
(
ϕ̂
)∣∣∣∣
p=0
∈ JJ (Ã1)
k−1,m,•.
Proof. A1-invariant. The vector field ∂
∂p in coordinates s, z1, z2, z3, τ reads
∂
∂p
=
∂
∂z1
+
∂
∂z2
+
∂
∂z3
+ 2g1(τ)p
∂
∂u
.
Moreover, in the coordinates s, z1, z2, z3, τ the A1 group acts by permuting z1 and z2. Then
∂
∂p
(ϕ(s, z2, z1, z3, τ))
∣∣∣∣
p=0
=
(
∂
∂z1
+
∂
∂z2
+
∂
∂z3
)
(ϕ(s, z2, z1, z3, τ))
∣∣∣∣
p=0
=
(
∂
∂z1
+
∂
∂z2
+
∂
∂z3
)
(ϕ(s, z1, z2, z3, τ))
∣∣∣∣
p=0
.
Translation invariant:
∂
∂p
(ϕ(u− 〈λ, v〉E − 〈λ, λ〉E , v + p+ λτ + µ, τ))
∣∣∣∣
p=0
=
∂
∂p
〈λ, v〉E
∣∣∣∣
p=0
ϕ(u, v, τ) +
∂ϕ
∂p
(
u− 〈λ, v〉Ã1
− 1
2
〈λ, λ〉Ã1
τ + k, v + λτ + µ, τ
)
=
∂ϕ
∂p
(
u− 〈λ, v〉Ã1
− 1
2
〈λ, λ〉Ã1
τ + k, v + λτ + µ, τ
)
=
∂ϕ
∂p
(u, v, τ).
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 15
SL2(Z)-equivariant of weight k:
∂
∂p
(
ϕ
(
u+
c〈v, v〉E
2(cτ + d)
,
v
cτ + d
,
p
cτ + d
,
aτ + b
cτ + d
)) ∣∣∣∣
p=0
=
c
2(cτ+ d)
∂
∂p
〈v, v〉E
∣∣∣∣
p=0
ϕ(u, v, τ)+
1
cτ+ d
∂ϕ
∂p
(
u+
c〈v, v〉E
2(cτ+ d)
,
v
cτ+ d
,
p
cτ+ d
,
aτ+ b
cτ+ d
)
=
1
cτ + d
∂ϕ
∂p
(
u+
c〈v, v〉E
2(cτ + d)
,
v
cτ + d
,
p
cτ + d
,
aτ + b
cτ + d
)
=
1
(cτ + d)k
∂ϕ
∂p
(u, v, τ).
Then,
∂ϕ
∂p
(
u+
c〈v, v〉E
2(cτ + d)
,
v
cτ + d
,
p
cτ + d
,
aτ + b
cτ + d
)
=
1
(cτ + d)k−1
∂ϕ
∂p
(u, v, τ).
Index 1:
1
2πi
∂
∂u
∂
∂p
ϕ̂ =
1
2πi
∂
∂p
∂
∂u
ϕ̂ =
∂
∂p
ϕ̂. �
Corollary 2.14. The function[
e
z ∂
∂p
(
e2πiu
θ1(v0 + v2 + p)θ1(−v0 + v2 + p)
θ1(2v2 + p)θ′1(0)
)] ∣∣∣∣
p=0
= ϕ
J (Ã1)
1 + ϕ
J (Ã1)
0 z +O(z2),
generates the Jacobi forms ϕ
J (Ã1)
0 and ϕ
J (A1)
1 , where
ϕ
J (Ã1)
0 :=
∂
∂p
(
ϕ̂
J (Ã1)
1
)∣∣∣∣
p=0
.
Proof. Acting ∂
∂p k times in ϕ
J (Ã1)
1 , we have
[
∂k
∂kp
(
e2πiu
θ1(v0 + v2 + p)θ1(−v0 + v2 + p)
θ1(2v2 + p)θ′1(0)
)] ∣∣∣∣
p=0
∈ JJ (Ã1)
1−k,1,•. �
Corollary 2.15. The generating function can be written as[
e
z ∂
∂p
(
e2πiu
θ1(v0 + v2 + p)θ1(−v0 + v2 + p)
θ1(2v2 + p)θ′1(0)
)] ∣∣∣∣
p=0
= e−2πi(u+ig1(τ)z2) θ1(z − v0 + v2, τ)θ1(z + v0 + v2, τ)
θ′1(0)θ1(z + 2v2)
.
Proof.[
e
z ∂
∂p
(
e2πiu
θ1(v0 + v2 + p)θ1(−v0 + v2 + p)
θ′1(0)θ1(2v2 + p)
)] ∣∣∣∣
p=0
=
[
e
z ∂
∂p
(
e−2πi(s+ig1(τ)p2) θ1(v0 + v2 + p)θ1(−v0 + v2 + p)
θ1(2v2 + p)θ′1(0)
)] ∣∣∣∣
p=0
= e−2πi(u+ig1(τ)z2) θ1(z − v0 + v2, τ)θ1(z + v0 + v2, τ)
θ′1(0)θ1(z + 2v2)
. �
16 G.F. Almeida
The next lemma is one of the main points of inquiry in this section, because this lemma
identify the orbit space of the group J
(
Ã1
)
with the Hurwitz space H1,0,0. This relationship
is possible due to the construction of the generating function of the Jacobi forms of type Ã1,
which can be completed to be the Landau–Ginzburg superpotential of H1,0,0 as follows:
e−2πi(u+ig1(τ)z2) θ1(z − v0 + v2, τ)θ1(z + v0 + v2, τ)
θ′1(0)θ1(z + 2v2)
7→ e−2πiu
θ1(v − v0 + v2, τ)θ1(v + v0 + v2, τ)
θ1(vτ)θ1(v + 2v2, τ)
.
Lemma 2.16. There exists a local isomorphism between Ω/J
(
Ã1
)
and H1,0,0.
Proof. The correspondence is realized by the map
[(u, v0, v2, τ)]←→ λ(v) = e−2πiu
θ1(v − v0, τ)θ1(v + v0, τ)
θ1(v − v2, τ)θ1(v + v2, τ)
, (2.19)
where θ1(v, τ) is the Jacobi θ1-function defined on (1.5).
It is necessary to prove that the map is well defined and one to one.
Well defined. Note that the map does not depend on the choice of the representative of
[(u, v0, v2, τ)] if the function (2.19) is invariant under the action of J
(
Ã1
)
. Therefore, let us
prove the invariance of the map (2.19).
A1-invariant. The A1 group acts on (2.19) by permuting its roots, thus (2.19) remains
invariant under this operation.
Translation invariant. Recall that under the translation v 7→ v+m+nτ , the Jacobi θ-function
transforms as [22]:
θ1(vi + µi + λiτ, τ) = (−1)λi+µie−2πi
(
λivi+
λ2i
2
τ
)
θ1(vi, τ). (2.20)
Then, substituting the transformation (2.20) into (2.19), we conclude that (2.19) remains inva-
riant.
SL2(Z)-invariant. Under SL2(Z)-action the following function transforms to
θ1
(
vi
cτ+d ,
aτ+d
cτ+d
)
θ′1
(
0, aτ+dcτ+d
) = (cτ + d)−1 exp
(
πicv2i
cτ + d
)
θ1(vi, τ)
θ′1(0, τ)
. (2.21)
Then, substituting the transformation (2.21) into (2.19), we conclude that (2.19) remains inva-
riant.
Injectivity. Note that for fixed v, v0, v2, u, the function τ 7→ f(τ) := λ(v, v0, v2, u, τ) is
a modular form with character [12]. This is clear because λ(v, v0, v2, u, τ) is rational function
of θ1(z, τ), which is modular form with character for special values of z [12]. If λ(v, v0, v2, u, τ) =
λ(v, v̂0, v̂2, û, τ̂), then for fixed v, v0, v2, u, v̂0, v̂2, û, we have f(τ) = f(τ̂); in particular,
f(τ), f(τ̂) have the same vanishing order, and this implies that τ , τ̂ belongs to the same SL2(Z)
orbit.
Two elliptic functions are equal if they have the same zeros and poles with multiplicity
mod Z⊕ τZ. So, for a fixed τ in the SL2(Z) orbit
v̂0 = v0 + λ0τ + µ0, v̂2 = v2 + λ2τ + µ2, (λi, µi) ∈ Z2.
Furthermore, for two different representations of the same SL2(Z) orbit, but considering fixed
cells, we have
v̂0 =
v0
cτ + d
, v̂2 =
v2
cτ + d
, τ̂ =
aτ + b
cτ + d
,
where
(
a b
c d
)
∈ SL2(Z).
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 17
Since, λ(v, v0, v2, u, τ) is invariant under translations, and SL2(Z), for τ̂ = τ , we have
û = u− 〈λ, v〉Ã1
− 〈λ, λ〉Ã1
τ
2
+ k.
For τ̂ = aτ+b
cτ+d ,
û = u−
c〈v, v〉Ã1
2(cτ + d)
+ k,
where k ∈ Z.
Surjectivity. Any elliptic function can be written as rational functions of Weierstrass σ-
function up to a multiplication factor [22]; by using the formula
σ(vi, τ) =
θ1(vi, τ)
θ′1(0, τ)
exp
(
−2πig1(τ)v2i
)
, g1(τ) =
η′(τ)
η(τ)
,
where η(τ) is the Dedekind η-function, we get the desire result. �
Remark 2.17. Lemma 2.16 is a local biholomorphism of manifolds, but this does not necessarily
means isomorphism of Dubrovin–Frobenius structure. On a Hurwitz space, there may exist in
several inequivalent Dubrovin–Frobenius structures. For instance, in [17] Romano constructed
two generalised WDDV solution on the Hurwitz space H1,0,0. Furthermore, in [2] and [3],
Bertola constructed two different Dubrovin–Frobenius structures on the orbit space of the Jacobi
group G2. The Dubrovin–Frobenius structure of this orbit space will be constructed only in
Section 3.
Remark 2.18. Lemma 2.16 associates a group to H1,0,0, and this could be useful for the general
understanding of the WDDV solutions/discrete group correspondence [6].
Corollary 2.19. The functions
(
ϕÃ1
0 , ϕÃ1
1
)
obtained by the formula
λÃ1 = e−2πiu
θ1(v − v0, τ)θ1(v + v0, τ)
θ1(v − v2, τ)θ1(v + v2, τ)
= ϕÃ1
1 [ζ(v − v2, τ)− ζ(v + v2, τ) + 2ζ(v2, τ)] + ϕÃ1
0 (2.22)
are Jacobi forms of weight 0, −1 respectively, index 1, and order 0. More explicitly,
ϕÃ1
1 =
θ1(v0 + v2, τ)θ1(−v0 + v2, τ)
θ′1(0, τ)θ1(2v2, τ)
e−2πiu,
ϕÃ1
0 = −ϕÃ1
1 [ζ(v0 − v2, τ)− ζ(v0 + v2, τ) + 2ζ(v2, τ)] , (2.23)
where ζ(v, τ) is the Weierstrass ζ-function for the lattice (1, τ), i.e.,
ζ(v, τ) =
1
v
+
∞∑
m2+n2 6=0
1
v −m− nτ
+
1
m+ nτ
+
v
(m+ nτ)2
.
Proof. Let us prove each item separately.
A1-invariant, translation invariant. The first line of (2.22) are A1-invariant, and translation
invariant by the lemma (2.16). Then, by the Laurent expansion of λÃ1 , we have that ϕÃ1
i are
A1-invariant, and translation invariant.
18 G.F. Almeida
SL2(Z)-equivariant. The first line of (2.22) are SL2(Z)-invariant, but the Weierstrass ζ-
functions of the second line of (2.22) have the following transformation law
ζ
(
z
cτ + d
,
aτ + b
cτ + d
)
= (cτ + d)ζ(z, τ).
Then, ϕÃ1
i must have the following transformation law:
ϕÃ1
0
(
u+
c〈v, v〉Ã1
2(cτ + d)
,
v
cτ + d
,
aτ + b
cτ + d
)
= ϕÃ1
0 (u, v, τ),
ϕÃ1
1
(
u+
c〈v, v〉Ã1
2(cτ + d)
,
v
cτ + d
,
aτ + b
cτ + d
)
= (cτ + d)−kϕÃ1
1 (u, v, τ).
Index 1:
1
2πi
∂
∂u
λÃ1 = λÃ1 .
Then
1
2πi
∂
∂u
ϕÃ1
i = ϕÃ1
i .
Analytic behavior. Note that λÃ1θ21(2v2, τ) is holomorphic function in all the variables vi.
Therefore, ϕÃ1
i are holomorphic functions on the variables v0, and meromorphic function in the
variable v2 with poles on j
2 + lτ
2 , j, l = 0, 1 of order 2, i.e., l = 0, since m = 1 for all ϕÃ1
i .
To prove the formula (2.23) let us compute the following limit:
lim
z→v2
λÃ1v2 = ϕÃ1
1 = e−2πiu
θ1(v0 + v2, τ)θ1(−v0 + v2, τ)
θ′1(0, τ)θ1(2v2, τ)
.
Let us also compute the zeros of λÃ1
λÃ1(v0) = 0 = ϕÃ1
1 [ζ(v0 − v2, τ)− ζ(v0 + v2, τ) + 2ζ(v2, τ)] + ϕÃ1
0 . �
Lemma 2.20. The functions ϕÃ1
0 , ϕÃ1
1 are algebraically independent over the ring E•,•.
Proof. If P (X,Y ) is any polynomial in E•,•(X,Y ), such that P
(
ϕÃ1
0 , ϕÃ1
1
)
= 0, then, the fact
ϕÃ1
0 , ϕÃ1
1 have an index that implies that each homogeneous component Pd
(
ϕÃ1
0 , ϕÃ1
1
)
has to
vanish identically. Defining pd
(
ϕ
Ã1
0
ϕ
Ã1
1
)
:=
Pd
(
ϕ
Ã1
0 ,ϕ
Ã1
1
)(
ϕ
(Ã1)
1
)d , we have that pd
(
ϕ
Ã1
0
ϕ
Ã1
1
)
is identically 0 iff
ϕ
Ã1
0
ϕ
Ã1
1
is constant (belongs to E•,•), but
ϕÃ1
0
ϕÃ1
1
=
℘′(v2, τ)
℘(v0, τ)− ℘(v2, τ)
6= a(v2, τ),
where a(v2, τ) is any function belongs to E•,•. Then, ϕÃ1
0 , ϕÃ1
1 are algebraically independent
over the ring E•,•.
Recall that ℘(v, τ) is the Weierstrass P-function (2.6). �
Consider the Landau–Ginzburg superpotential (2.24) for the J (A2) case below.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 19
Theorem 2.21 ([2]). The ring of A2-invariant Jacobi forms is free module of rank 3 over the
ring of modular forms, moreover there exist a formula for its generators given by
λA2 = e−2πiu2
θ1(z + v0 + v2, τ)θ1(z − v0 + v2, τ)θ1(z − 2v2)
θ31(z, τ)
= −ϕ
A2
3
2
℘′(z, τ) + ϕA2
2 ℘(z, τ) + ϕA2
0 . (2.24)
Lemma 2.22. Let
{
ϕÃ1
0 , ϕÃ1
1
}
be set of functions given by the formula (2.22), and{
ϕA2
0 , ϕA2
2 , ϕA2
3
}
given by (2.24), then
ϕA2
3 = ϕÃ1
1 ϕA1
2 ,
ϕA2
2 = ϕÃn0 ϕA1
2 + a2(v2, τ)ϕÃnj ϕA1
2 ,
ϕA2
0 = a0(v2, τ)ϕÃ1
0 ϕA1
2 + b0(v2, τ)ϕÃ1
2 ϕA1
2 ,
where
ϕA1
2 :=
θ21(2v2, τ)
θ′1(0, τ)2
e2πi(−u2+u1)
and ai, bi are elliptic functions on v2.
Proof. Note the following relation
λA2
λÃ1
=
θ1(z − 2v2, τ)θ1(z + 2v2), τ
θ21(z, τ)
e2πi(−u2+u1) = ϕA1
2 ℘(z, τ)− ϕA1
2 ℘(2v2, τ).
Hence,
−ϕ
A2
3
2
℘′(z, τ) + ϕA2
2 ℘(z, τ) + ϕA2
0 =
(
ϕÃ1
1 [ζ(z, τ)− ζ(z + 2v2, τ) + 2ζ(v2, τ)] + ϕÃn0
)
×
(
ϕA1
2 ℘(z, τ)− ϕA1
2 ℘(2v2, τ)
)
.
Then, the desired result is obtained by doing a Laurent expansion in the variable z on both
sides of the equality. �
Corollary 2.23.
E•,•
[
ϕÃ1
0 , ϕÃ1
1
]
= E•,•
[
ϕA2
0
ϕA1
2
,
ϕA2
2
ϕA1
2
,
ϕA2
3
ϕA1
2
]
.
Moreover, we have the following lemma:
Lemma 2.24. Let be ϕ ∈ J Ã1
•,•,m, then ϕ ∈ E•,•
[
ϕ
A2
0
ϕ
A1
2
,
ϕ
A2
2
ϕ
A1
2
,
ϕ
A2
3
ϕ
A1
2
]
.
Proof. Let be ϕ ∈ J Ã1
•,•,m, then the function ϕ(
ϕ
Ã1
1
)m is an elliptic function on the variables
(v0, v2) with poles on v0 − v2, v0 + v2, 2v2 due to the zeros of ϕÃ1
1 and the poles of ϕ, which are
by definition in 2v2. Expanding the function ϕ(
ϕ
Ã1
1
)m in the variables v0, v2 we get
ϕ(
ϕÃ1
1
)m =
m∑
i=−1
ai℘(i)(v0 + v2) +
m∑
i=−1
bi℘(i)(−v0 + v2) + c(v2, τ), (2.25)
where ℘−1(v) := ζ(v), and c(v2, τ) is an elliptic function in the variable v2.
20 G.F. Almeida
However, the function ϕ(
ϕ
Ã1
1
)m is invariant under the permutations of the variables v0, so the
equation (2.25) is
ϕ(
ϕÃ1
1
)m =
m∑
i=−1
ai
(
℘(i)(v0 + v2) + ℘(i)(−v0 + v2)
)
+ c(v2, τ). (2.26)
Now we complete this function to A2-invariant function by summing and subtracting the
following function in equation (2.26)
f(v2, τ) =
m∑
i=−1
ai℘(i)(2v2).
Hence,
ϕ(
ϕÃ1
1
)m =
m∑
i=−1
ai
(
℘(i)(v0 + v2) + ℘(i)(−v0 + v2) + ℘(i)(2v2)
)
+ g(v2, τ). (2.27)
Multiplying both side of the equation (2.27) by ϕA1
1 , we get
ϕ =
(
m∑
i=−1
ai
(
℘(i)(v0 + v2) + ℘(i)(−v0 + v2) + ℘(i)(2v2)
)) (
ϕA2
3
)m
+ g(v2, τ)
(
ϕA2
3
)m
.
To finish the proof, we will show that(
m∑
i=−1
ai
(
℘(i)(v0 + v2) + ℘(i)(−v0 + v2) + ℘(i)(2v2)
)) (
ϕA2
3
)m
is a weak holomorphic Jacobi form of type A2. To finish the proof, note the following:
1. The functions(
ϕA2
3
)m(
℘(i)(v0 + v2) + ℘(i)(−v0 + v2) + ℘(i)(2v2)
)
(2.28)
are A2-invariant by construction.
2. The functions (2.28) are invariant under the action of (Z⊕ τZ)2, because ϕA2
3 is invariant,
and
℘(i)(v0 + v2) + ℘(i)(−v0 + v2) + ℘(i)(2v2) (2.29)
are elliptic functions.
3. The functions (2.28) are equivariant under the action of SL2(Z), because ϕA2
3 is equivariant,
and (2.29) are elliptic functions.
4. The function ϕA2
3 has zeros on v0 − v2, v0 + v2, 2v2 of order m, and (2.29) has poles
on v0 − v2, v0 + v2, 2v2 of order i+ 2 ≤ m. So, the functions (2.28) are holomorphic.
Hence,
ϕ ∈ E•,•
[
ϕA2
0
ϕA1
2
,
ϕA2
2
ϕA1
2
,
ϕA2
3
ϕA1
2
]
. �
At this stage, the principal theorem can be stated in a precise way as follows.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 21
Theorem 2.25. The trigraded algebra of Jacobi forms J
J (Ã1)
•,•,• =
⊕
k,l,m J
Ã1
k,l,m is freely generated
by 2 fundamental Jacobi forms
(
ϕÃ1
0 , ϕÃ1
1
)
over the graded ring E•,•
J
J (Ã1)
•,•,• = E•,•
[
ϕÃ1
0 , ϕÃ1
1
]
.
Proof.
J Ã1
•,•,• ⊂ E•,•
[
ϕA2
0
ϕA1
2
,
ϕA2
2
ϕA1
2
,
ϕA2
3
ϕA1
2
]
= E•,•
[
ϕÃ1
0 , ϕÃ1
1
]
⊂ J Ã1
•,•,•. �
Remark 2.26. The structural difference between the Chevalley theorems of the groups J(A1)
and J
(
Ã1
)
lies in the ring of coefficients. The ring of coefficients of Jacobi forms with respect to
J(A1) are modular forms, and the ring of coefficients of Jacobi forms with respect to J
(
Ã1
)
are
the ring of elliptic functions with poles on 0, 12 ,
τ
2 ,
1+τ
2 mod Z⊕τZ, for fixed τ . See Lemma 2.11.
Remark 2.27. The geometry of ΩJ (Ã1)/J
(
Ã1
)
is similar to ΩJ (A1)/J (A1). Indeed, the orbit
space of J
(
Ã1
)
is locally a line bundle over a family of two elliptic curves, Eτ/A1 ⊗ Eτ , where
the first one is quotient by A1, and both are parametrised by H/SL2(Z).
3 Frobenius structure on the orbit space of J
(
Ã1
)
In this section, a Dubrovin–Frobenius manifold structure will be constructed on the orbit space
of J
(
Ã1
)
. More precisely, it will define the data
(
ΩJ (Ã1/J
(
Ã1
)
, g∗, e, E
)
, with the intersection
form g∗, unit vector field e, and Euler vector field E. This data will be written naturally in
terms of the invariant functions of J
(
Ã1
)
. Thereafter, it will be proved that this data is enough
to the construction of the Dubrovin–Frobenius structure.
3.1 Intersection form
The first step to be done is to construct the intersection form. It will be shown that such metric
can be constructed by using just the data of the group J
(
Ã1
)
. The strategy is to combine
the intersection form of the group Ã1 and J (A1). Recall that the intersection form of the
group Ã1 [6, 10] is
ds2 = 2dv20 − 2dv22,
and the intersection form of J (A1) [2, 3, 6] is
ds2 = dv20 + 2dudτ.
Therefore, the natural candidate to be the intersection form of J
(
Ã1
)
is
ds2 = 2dv20 − 2dv22 + 2dudτ.
The following lemma proves that this metric is invariant metric of the group J
(
Ã1
)
. To be
precise, the metric will be invariant under the action of A1, and translations, and equivariant
under the action of SL2(Z).
Lemma 3.1. The metric
ds2 = 2dv20 − 2dv22 + 2dudτ (3.1)
is invariant under the transformations (2.9), (2.10). Moreover, the transformations (2.11) deter-
mine a conformal transformation of the metric ds2, i.e.,
2dv20 − 2dv22 + 2dudτ 7→ 2dv20 − 2dv22 + 2dudτ
(cτ + d)2
.
22 G.F. Almeida
Proof. Under (2.9), (2.10), the differentials transform to
dv0 7→ −dv0, dv0 7→ dv0 + λ0dτ, dv2 7→ dv2 + λ2dτ,
du 7→ du− λ20dτ − 2λ0dv0 + λ22dτ + 2λ2dv2, dτ 7→ dτ.
Hence,
dv20 7→ dv20, dv20 7→ dv20 + 2λ0dv0dτ + λ20dτ
2, dv22 7→ dv22 + 2λ2dv2dτ + λ22dτ
2,
2dudτ 7→ 2dudτ − 2λ20dτ
2 − 4λ0dv0dτ + 2λ22dτ
2 + 4λ2dv2dτ.
So,
2dv20 − 2dv22 + 2dudτ 7→ 2dv20 − 2dv22 + 2dudτ.
Let us show that the metric has conformal transformation under the transformations (2.11)
dv0 7→
dv0
cτ + d
− v0dτ
(cτ + d)2
, dv2 7→
dv2
cτ + d
− v2dτ
(cτ + d)2
, dτ 7→ dτ
(cτ + d)2
,
du 7→ du+
c(2v0dv0 − 2v2dv
2
2)
cτ + d
− c(v20 − v22)dτ
(cτ + d)2
.
Then,
dv20 7→
dv20
(cτ + d)2
− 2v0dv0dτ
(cτ + d)3
+
v20dτ2
(cτ + d)4
, dv22 7→
dv22
(cτ + d)2
− 2v2dv2dτ
(cτ + d)3
+
v22dτ2
(cτ + d)4
,
2dudτ 7→ 2dudτ
(cτ + d)2
+
c(4v0dv0 − 4v2dv2)dτ
(cτ + d)3
− c(2v20 − 2v22)dτ2
(cτ + d)4
.
Then,
2dv20 − 2dv22 + 2dudτ 7→ 2dv20 − 2dv22 + 2dudτ
(cτ + d)2
. �
3.2 Euler and unit vector field
The next step is to construct a two vector field, which is a intrinsic object of the orbit spa-
ce J
(
Ã1
)
. The first one is the Euler vector
E = − 1
2πi
∂
∂u
, (3.2)
which was already defined in the last equation of (2.14). Therefore, it is an already intrinsic
object, since it comes from the definition of meromorphic Jacobi forms associated with J
(
Ã1
)
.
In the invariant coordinates, the vector field (3.2) reads as
E = ϕ0
∂
∂ϕ0
+ ϕ1
∂
∂ϕ1
.
The second one is given by the coordinates (ϕ0, ϕ1, v2, τ) as
e =
∂
∂ϕ0
, (3.3)
and it is denoted by the unit vector field. This object is intrinsic to the orbit space of J
(
Ã1
)
,
because it is written in terms of a meromorphic Jacobi forms associated with J
(
Ã1
)
.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 23
3.3 Flat coordinates of the Saito metric
In order to construct the Dubrovin–Frobenius structure, it will be necessary to introduce the
coordinates
(
t1, t2, t3, t4
)
.
Lemma 3.2. There is a change of coordinates in ΩJ (Ã1)/J
(
Ã1
)
, given by
t1 = ϕ0 + 2t2
θ′1(v2|τ)
θ1(v2|τ)
, t2 = ϕ1, t3 = v2, t4 = τ.
Proof. Note that the function (2.19) can be parametrised by
(
t1, t2, t3, t4
)
as follows:
λ = ϕ0 + ϕ1[ζ(v − v2|τ)− ζ(v + v2|τ) + 2ζ(v2)]
= ϕ0 + ϕ1
[
θ′1(v − v2|τ)
θ1(v − v2|τ)
− θ′1(v + v2|τ)
θ1(v + v2|τ)
+ 2
θ′1(v2|τ)
θ1(v2|τ)
]
= ϕ0 + 2
θ′1(v2|τ)
θ1(v2|τ)
+ ϕ1
[
θ′1(v − v2|τ)
θ1(v − v2|τ)
− θ′1(v + v2|τ)
θ1(v + v2|τ)
]
= t1 + t2
[
θ′1
(
v − t3|t4
)
θ1
(
v − t3|t4
) − θ′1
(
v + t3|t4
)
θ1
(
v + t3|t4
)]
from the first line to the second line, the following equation was used:
ζ(v − v2, τ) =
θ′1(v − v2|τ)
θ1(v − v2|τ)
+ 4πig1(τ)(v − v2).
In this way,
(
t1, t2, t3, t4
)
are local coordinates of ΩJ (Ã1)/J
(
Ã1
)
due to Lemma 2.16. �
The side back effect of the coordinates
(
t1, t2, t3, t4
)
is the fact that they are not globally
single valued functions on the quotient.
Lemma 3.3. The coordinates
(
t1, t2, t3, t4
)
have the following transformation laws under the ac-
tion of the group J
(
Ã1
)
: they are invariant under (2.9). They transform as follows under (2.10):
t1 7→ t1 − λ2t2, t2 7→ t2, t3 7→ t3 + µ2 + λ2t
4, t4 7→ t4.
Moreover, they transform as follows under (2.11)
t1 7→ t1 +
2ct2t3
ct4 + d
, t2 7→ t2
ct4 + d
, t3 7→ t3
ct4 + d
, t4 7→ at4 + b
ct4 + d
.
Proof. The invariance under (2.9) is clear, since only t1 depends on v0, and its dependence is
given by ϕ0, which is invariant under (2.9). Let us check how tα transforms under (2.10), (2.11):
Since t3 = v2, t
4 = τ , we have the desired transformations law defined as J
(
Ã1
)
. The coordinate
t2 = ϕ1 is a invariant under (2.10) and transforms as modular form of weight −1 under (2.10).
The only non-trivial term is t1, because it contains the term
θ′1(v2|τ)
θ1(v2|τ) , which transforms as follows
under (2.10), (2.11) [22]
θ′1(v2|τ)
θ1(v2|τ)
7→ θ′1(v2|τ)
θ1(v2|τ)
− 2πin2,
θ′1(v2|τ)
θ1(v2|τ)
7→ (cτ + d)
θ′1(v2|τ)
θ1(v2|τ)
+ 2πict3.
The proof is completed when we do the rescaling from t1 to t1
2πi . �
24 G.F. Almeida
In order to make the coordinates
(
t1, t2, t3, t4
)
being well defined, it will be necessary to define
them in a suitable covering over ΩJ (Ã1)/J
(
Ã1
)
. It is clear that the multivaluedness comes from
the coordinates t3, t4 essentially. Therefore, the problem is solved by defining a suitable covering
over the orbit space of J
(
Ã1
)
. This can be done by fixing a lattice
(
1, t4
)
and a representative
of orbit given by the action
t3 7→ t3 + µ2 + λ2t
4. (3.4)
In order to also realise the coordinates (u, v0, v2, τ) as globally well-behaviour in the covering
of the orbit space of J
(
Ã1
)
, we also forget the A1-action by fixing a representative of each orbit.
Therefore, in the following covering the problem
˜
ΩJ (Ã1)/J
(
Ã1
)
:= ΩJ (Ã1)/Z⊕ τZ, (3.5)
where Z⊕ τZ acts on ΩJ (Ã1) as
v0 7→ v0 + λ0τ + µ0, u 7→ u− 2λ0v0 − n20τ, v2 7→ v2, τ 7→ τ.
In the covering (3.5) the coordinates tα, and the intersection form g∗ are globally single valued.
Hence, we have the necessary conditions to have Dubrovin–Frobenius manifold, since its geom-
etry structure should be globally well defined. Note that, ΩJ (Ã1)/J
(
Ã1
)
has the structure of
Twisted Frobenius manifold [6].
Remark 3.4.
(
t1, t2
)
lives in an enlargement of the algebra of E•,•[ϕ0, ϕ1]. The extended
algebra is the same as E•,•[ϕ0, ϕ1], but it is necessary to add the function
θ′1(v2,τ)
θ1(v2,τ)
in the ring
of coefficients E•,•.
Remark 3.5. Note that a covering in the orbit space corresponds to a covering in the Hurwitz
space. The fixation of a lattice in the orbit space of J
(
Ã1
)
is equivalent to a choice of homology
basis in the Hurwitz space H1,0,0. Moreover, a choice of the representative of the action (3.4)
in the variable v2 is a choice of logarithm root in the Hurwitz space H1,0,0. Furthermore, fixing
a representative of the A1-action is to choice a pole or equivalently to choice a sheet in the
Hurwitz space H1,0,0.
Remark 3.6. The Dubrovin–Frobenius structure in a Hurwitz space is based on an open dense
domain of a solution of a Darboux–Egoroff system [6, 19]. Hence, it is a local construction.
Indeed, the canonical coordinates associated to the Hurwitz spaces are local coordinates even
in the covering space described in Remark 3.5. The construction of the orbit space of J
(
Ã1
)
complements the construction of the Hurwitz space H1,0,0, because now, there exists global
object where the local Dubrovin–Frobenius structure of H1,0,0 lives. Indeed, the coordinates
(ϕ0, ϕ1, v2, τ) are global coordinates for the covering space (3.5), and from this fact, we derive
that the Dubrovin–Frobenius structure is globally well defined in the covering. This is possible
because we realise the group J
(
Ã1
)
as a monodromy of orbit space J
(
Ã1
)
, and we know how
the group J
(
Ã1
)
acts on the Dubrovin–Frobenius structure.
3.4 Construction of WDVV solution
Theorem 3.7. There exists Dubrovin–Frobenius structure on the manifold ˜Ω/J
(
Ã1
)
with the
intersection form (3.1), the Euler vector field (3.2), and the unity vector field (3.3). Moreover,
˜Ω/J
(
Ã1
)
is isomorphic as Dubrovin–Frobenius manifold to H̃1,0,0.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 25
Proof. The first step to be done is the computation of the intersection form in coordinates(
t1, t2, t3, t4
)
. Hence, consider the transformation formula of ds2:
gαβ(t) =
∂tα
∂xi
∂tβ
∂xj
gij , (3.6)
where x1 = u, x2 = v0, x
3 = v2, x
4 = τ .
From the expression:
ds2 = 2dv20 − 2dv22 + 2dudτ = gijdx
idxj ,
we have
(gij) =
0 0 0 1
0 2 0 0
0 0 −2 0
1 0 0 0
.
Therefore,
(
gij
)
= (gij)
−1 =
0 0 0 1
0 1
2 0 0
0 0 −1
2 0
1 0 0 0
.
To compute gαβ(t), let us write tα in terms of xi:
t4 = τ, t3 = v2, t2 = −θ1(v0 + v2, τ)θ1(v0 − v2, τ)
θ1(2v2, τ)θ′1(0, τ)
e−2πiu,
using the following formulae [22]:
℘′(v2)
℘(v0)− ℘(v0)
= ζ(v0 − v2, τ)− ζ(v0 + v2, τ) + 2ζ(v2, τ),
℘(v0, τ)− ℘(v2, τ) = −σ(v0 + v2, τ)σ(v0 − v2, τ)
σ2(v0, τ)σ2(v2, τ)
,
σ(2v2, τ)
σ4(v2, τ)
= −℘′(v2, τ),
it is possible to rewrite t1 in a more suitable way
t1 = −t2[ζ(v0 − v2, τ)− ζ(v0 + v2, τ) + 2ζ(v2, τ)] + 2t2
θ′1(v2, τ)
θ1(v2, τ)
= −t2 ℘′(v2, τ)
℘(v0, τ)− ℘(v2, τ)
+ 2t2
θ′1(v2, τ)
θ1(v2, τ)
= −t2 ℘′(v2, τ)θ21(v2, τ)θ21(v0, τ)
θ1(v0 + v2, τ)θ1(v0 − v2, τ)θ′1(0, τ)2
+ 2t2
θ′1(v2, τ)
θ1(v2, τ)
= −℘
′(v2, τ)θ21(v2, τ)θ21(v0, τ)
θ1(2v2, τ)θ′1(0, τ)3
e−2πiu + 2t2
θ′1(v2, τ)
θ1(v2, τ)
=
θ21(v0, τ)
θ21(v2, τ)
e−2πiu + 2t2
θ′1(v2, τ)
θ1(v2, τ)
.
To summarize
t1 =
θ21(v0, τ)
θ21(v2, τ)
e−2πiu + 2t2
θ′1(v2, τ)
θ1(v2, τ)
,
t2 = −θ1(v0 + v2, τ)θ1(v0 − v2, τ)
θ1(2v2, τ)θ′1(0, τ)
e−2πiu, t3 = v2, t4 = τ.
26 G.F. Almeida
Computing gαβ according to (3.6)
gαβ =
1
2
∂tα
∂v0
∂tβ
∂v0
− 1
2
∂tα
∂v2
∂tβ
∂v2
+
∂tα
∂u
∂tβ
∂τ
+
∂tα
∂τ
∂tβ
∂u
.
Trivially, we get
g44 = g34 = 0, g33 = −1
2
, g24 = −2πit2, g14 = −2πit1.
The following non-trivial terms are computed in Appendix A:
g23 = − t
1
2
+ t2
θ′1
(
2t3, τ
)
θ1
(
2t3, τ
) , g13 = −2πit2
∂
∂τ
(
log
θ′1(0, τ)
θ1
(
2t3, τ
)) ,
g22 = 2
(
t2
)2 [θ′′1(2t3, τ)
θ1(2t3, τ)
−
θ′
2
1
(
2t3, τ
)
θ21
(
2t3, τ
) ] ,
g12 = −2πi
(
t2
)2 [ ∂2
∂t3∂τ
(
log
(
θ′1(0, τ)
θ1
(
2t3, τ
)))] , (3.7)
g11 = −4
(
t2
)2 θ′1(t3, τ)
θ1
(
t3, τ
) ∂
∂t3
(
θ′1
(
t3, τ
)
θ1
(
t3, τ
))[2
θ′1
(
t3, τ
)
θ1
(
t3, τ
) − 2
θ′1
(
2t3, τ
)
θ1
(
2t3, τ
)]
+ 8
θ′
2
1
(
t3, τ
)
θ21
(
t3, τ
) (t2)2 [θ′′1(2t3, τ)
θ1
(
2t3, τ
) − θ′
2
1
(
2t3, τ
)
θ21
(
2t3, τ
) ]− 2
(
t2
)2 [ ∂
∂t3
(
θ′1
(
t3, τ
)
θ1
(
t3, τ
))]2
− 16πi
(
t2
)2 θ′1(t3, τ)
θ1
(
t3, τ
) ∂
∂τ
(
θ′1
(
t3, τ
)
θ1
(
t3, τ
)) . (3.8)
Differentiating gαβ w.r.t. t1 we obtain a constant matrix η∗:
(
ηαβ
)
=
∂
∂t1
(
gαβ
)
=
0 0 0 −2πi
0 0 −1
2 0
0 −1
2 0 0
−2πi 0 0 0
.
So t1, t2, t3, t4 are the flat coordinates.
The next step is to calculate the matrix Fαβ using the formula
Fαβ =
gαβ
deg
(
gαβ
) . (3.9)
We can compute deg
(
gαβ
)
using the fact that we compute deg(tα). Indeed,
E = − 1
2πi
∂
∂u
.
This implies that
deg
(
t1
)
= deg
(
t2
)
= 1, deg
(
t3
)
= deg
(
t4
)
= 0.
Then, the function F is obtained from the equation
∂2F
∂tα∂tβ
= ηαα′ηββ′Fα
′β′
.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 27
Computing
Fα4 =
gα4
deg
(
gα4
) ,
we derive
∂3F
∂t1∂tα∂tβ
= ηαβ.
Hence,
F =
i
4π
(
t1
)2
t4 − 2t1t2t3 + f
(
t2, t3, t4
)
. (3.10)
Substituting F 23 and F 13 in (3.10)
F =
i
4π
(
t1
)2
t4 − 2t1t2t3 −
(
t2
)2
log
(
θ′1
(
0, t4
)
θ1
(
2t3, t4
))+ h
(
t2
)
+Aαβt
αtβ + Cαt
α +D,
where Aαβ, Cα, Cα are constants. Note that F 22, F 12 contains the same information, further-
more, there is no information in F 33, F 34, F 44 because
deg
(
g33
)
= deg
(
g34
)
= deg
(
g44
)
= 0.
However, h
(
t2
)
can be computed by using g33
g33 = −1
2
= Eεη3µη3λcεµλ =
t2
4
c222.
Using the formula (1.1), we have
F
(
t1, t2, t3, t4
)
=
i
4π
(
t1
)2
t4 − 2t1t2t3 −
(
t2
)2
log
(
t2
θ′1
(
0, t4
)
θ1
(
2t3, t4
)) . (3.11)
The remaining part of proof is to show that the equation (3.11) satisfies WDDV equations. Let
us prove it step by step.
1. Commutative of the algebra. Defining the structure constant of the algebra as
cαβγ(t) =
∂3F
∂tα∂tβ∂tγ
,
commutative is straightforward.
2. Normalization. Using equation (3.11), we obtain
c1αβ(t) =
∂3F
∂t1∂tβ∂tγ
= ηαβ.
3. Quasi homogeneity. Applying the Euler vector field in the function (3.11), we have
E(F ) = 2F − 2t2.
4. Associativity. In order to prove that the algebra is associativity, we will first shown that
the algebra is semisimple. First of all, note that the multiplication by the Euler vector field is
equivalent to the intersection form. Indeed,
E • ∂α = tσcβσα∂β = tσ∂σ
(
ηβµ∂α∂µF
)
∂β = (dα − dβ)ηβµ∂α∂µF∂β = ηαµg
µβ∂β. (3.12)
28 G.F. Almeida
Therefore, the multiplication by the Euler vector field is semisimple if the following polynomial
det
(
ηαµg
µβ − uδβα
)
= 0, (3.13)
has only simple roots; since det(ηαµ) 6= 0, the equation (3.13) is equivalent to
det
(
gαβ − uηαβ
)
= 0.
Using that ηαβ = ∂1g
αβ, we have that
det
(
gαβ − uηαβ
)
= det
(
gαβ
(
t1 − u, t2, t3, t4
))
= 0.
So, this is enough to compute det gαβ. In particular, computing det g in the coordinates
(ϕ0, ϕ1, v2, τ). Recall that
gαβ = g
(
dtα, dtβ
)
, in coordinates
(
t1, t2, t3, t4
)
,
glm = g(dvl,dvm), in coordinates (u, v0, v2, τ),
gij = g(dϕi,dϕj), in coordinates (ϕ0, ϕ1, v2, τ).
Then,
det gij = det
(
∂ϕi
∂vl
)
det
(
∂ϕj
∂vm
)
det
(
glm
)
.
Remark 3.8. The coordinates (u, v0, v2, τ) are defined away from the submanifold defined by
det g = 0. Therefore, we have to change coordinates to compute the roots of det g = 0.
Hence, it is enough to compute the det
(∂ϕi
∂vl
)
det
(
∂ϕi
∂vl
)
=
∂ϕ0
∂v0
∂ϕ0
∂v2
∂ϕ0
∂τ
−2πiϕ0
∂ϕ1
∂v0
∂ϕ1
∂v2
∂ϕ1
∂τ
−2πiϕ1
0 1 0 0
0 0 1 0
= −2πiϕ0ϕ1
[
2
θ′1(v0)
θ1(v0)
− θ′1(−v0 + v2)
θ1(−v0 + v2)
+
θ′1(v0 + v2)
θ1(v0 + v2)
]
= −2πie−4πiu
θ1(2v0)
θ1(2v2)θ′1(0)2
. (3.14)
Then, equation (3.14) has four distinct roots v0 = 0, 12 ,
τ
2 ,
1+τ
2 . Hence, the following system of
equations
det
(
gαβ
(
t1, t2, t3, t4
))
= 0, det
(
ηαβ
(
t1, t2, t3, t4
))
6= 0, (3.15)
implies in existence of four functions yi
(
t2, t3, t4
)
such that
t1 = yi
(
t2, t3, t4
)
, i = 1, 2, 3, 4.
Sending t1 7→ t1 − u in (3.15), we obtain
ui = t1 − yi
(
t2, t3, t4
)
, i = 1, 2, 3, 4.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 29
The multiplication by the Euler vector field
gij = ηjkg
ki, in canonical coordinates
(
u1, u2, u3, u4
)
is diagonal, so
gij = uiηijδij ,
where ηij is the canonical coordinates
(
u1, u2, u3, u4
)
, and the unit vector field have the following
form:
∂
∂t1
=
4∑
i=1
∂ui
∂t1
∂
∂ui
=
4∑
i=1
∂
∂ui
.
Moreover, since
[E, e] =
[
t1
∂
∂t1
+ t2
∂
∂t2
,
∂
∂t1
]
= −e,
the Euler vector field in the coordinates
(
u1, u2, u3, u4
)
takes the following form:
E =
4∑
i=1
ui
∂
∂ui
.
Using the relationship (3.12) between the coordinates
(
u1, u2, u3, u4
)
, we have
uiηijδij = ulηimηjnclmn, (3.16)
differentiating both side of the equation (3.16) with respect t1
ckij = δij ,
which proves that the algebra is associative and semisimple.
Therefore, we proved that the equation (3.11) satisfies the WDVV equation. Moreover, the
function F (3.11) is exactly the Free energy of the Dubrovin–Frobenius manifold of the Hurwitz
space H̃1,0,0. Hence, the covering of orbit space of J
(
Ã1
)
and the covering over the Hurwtiz
space H1,0,0 are isomorphic as a Dubrovin–Frobenius manifold, because they have the same
WDVV solution. �
Remark 3.9. Even thought the Dubrovin–Frobenius structure constructed in a suitable cover-
ing of the orbit space of J
(
Ã1
)
is isomorphic as a Dubrovin–Frobenius manifold to a suitable
covering of the Hurwitz space H1,0,0, this fact does not mean that the construction presented
in this paper is equivalent to the Hurwitz space construction, because:
1. The constructions start with different hypotheses. Indeed, we derive a WDVV solution
in the Hurwitz space framework from the data of the Hurwitz space itself, and through
the choice of a suitable primary differential; see [6] and [19] for the definition. On another
hand, the orbit space construction is derived from the data of the group J
(
Ã1
)
.
2. The Hurwitz space construction is based on domain of a solution of a Darboux–Egorrof
system. The coordinate system associated with this system of equation is called canonical
coordinates. Therefore, the Hurwitz space construction is a local construction, since it is
based in a local solution of a system of equations. The orbit space construction, in the
other hand, is built based on the invariant coordinates (ϕ0, ϕ1, v2, τ), which some how
have a global meaning. Furthermore, note that the existence of the invariant coordinates
(ϕ0, ϕ1, v2, τ) is not guaranteed in the Hurwitz space construction.
30 G.F. Almeida
3. The intersection form (3.1), the Euler vector field (3.2), and the unity vector field (3.3)
are intrinsic objects of the orbit space J
(
Ã1
)
. Therefore, the Theorem 3.7 derives the
WDVV solution (3.11) by using the equation (3.9) without using the correspondence with
the Hurwtiz space H1,0,0. This argument was already used in the introduction of [7] to
demonstrate the difference between the Hurwitz space construction on the H0,n and the
orbit space construction of the orbit space of An.
Remark 3.10. The WDVV solution (3.11) was presented on p. 28 of [8]. However, there is
a typo in the last term of the WDVV solution in the paper [8]. The WDVV solution in a correct
form can also be found in [5] and [13].
4 Conclusion
The WDVV solution of H1,0,0, which is (3.11), contains the term log
( θ′1(0,t
4)
θ1(2t3,t4)
)
on the two
exceptional variables
(
t3, t4
)
. This is a reflection of how the ring of invariants affects the WDVV
solution. The same pattern is obtained in J (A1), and Ã1. The equation (2.8) contains E2(τ)
which is a quasi modular form, and the equation (2.2) contains et
2
. These facts could be useful
in regards to the understanding of the WDVV/groups correspondence.
The arrows of the diagram of in Section 2.1 may have a third meaning, which is an embedding
of Dubrovin–Frobenius submanifolds [20, 21] in to the ambient space H1,0,0. The fact that H1,0,0
contains three Dubrovin–Frobenius submanifolds is not an accident. This comes from the tri-
Hamiltonian structure that H1,0,0 has [15, 16]. In a subsequent publication, we will study the
Dubrovin–Frobenius manifolds of H1,0,0, and its associated integrable systems.
A Appendix
Computing g12:
g23 = −1
2
∂t2
∂v2
= − t
2
2
[
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
= − t
2
2
[
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(v2, τ)
θ1(v2, τ)
]
− t2 θ
′
1(v2, τ)
θ1(v2, τ)
+ t2
θ′1(2v2, τ)
θ1(2v2, τ)
= − 1
2℘′(v2, τ)
[−ζ(v0 − v2, τ) + ζ(v0 + v2, τ)− 2ζ(v2, τ)]− t2 θ
′
1(v2, τ)
θ1(v2, τ)
+ t2
θ′1(2v2, τ)
θ1(2v2, τ)
=
1
2
1
℘(z0, τ)− ℘(z2, τ)
− t2 θ
′
1(v2, τ)
θ1(v2, τ)
+ t2
θ′1(2v2, τ)
θ1(2v2, τ)
= − t
1
2
+ t2
θ′1(2v2, τ)
θ1(2v2, τ)
.
Computing g13:
g13 = −1
2
∂t1
∂v2
= −θ
′
1(v2, τ)
θ1(v2, τ)
1
℘(z0)− ℘(z2)
− ∂t2
∂v2
θ′1(v2, τ)
θ1(v2, τ)
− t2
[
θ′′1(v, τ)
θ1(v, τ)
− θ′
2
1 (v, τ)
θ21(v, τ)
]
= −θ
′
1(v2, τ)
θ1(v2, τ)
1
℘(z0)− ℘(z2)
− t1 θ
′
1(v2, τ)
θ1(v2, τ)
− 2t2
θ′1(2v2, τ)
θ1(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
− t2
[
θ′′1(v, τ)
θ1(v, τ)
− θ′
2
1 (v, τ)
θ21(v, τ)
]
= −2t2
θ′
2
1 (v2, τ)
θ21(v2, τ)
− 2t2
θ′1(2v2, τ)
θ1(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
− t2
[
θ′′1(v, τ)
θ1(v, τ)
− θ′
2
1 (v, τ)
θ21(v, τ)
]
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 31
= −t2 θ
′2
1 (v2, τ)
θ21(v2, τ)
− 2t2
θ′1(2v2, τ)
θ1(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
− t2 θ
′′
1(v, τ)
θ1(v, τ)
. (A.1)
To simplify this expression we need the following lemma.
Lemma A.1 ([2]). When x+ y + z = 0 holds
θ′′1(x, τ)
θ1(x, τ)
+
θ′′1(y, τ)
θ1(y, τ)
− 2
θ′1(x, τ)
θ1(x, τ)
θ′1(y, τ)
θ1(y, τ)
= 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(x− y, τ)
))
+ 2
θ′1(x− y, τ)
θ1(x− y, τ)
[
θ′1(x, τ)
θ1(x, τ)
− θ′1(y, τ)
θ1(y, τ)
]
. (A.2)
Proof. Applying the formulas
ζ(v, τ) =
θ′1(v, τ)
θ1(v, τ)
+ 4πig1(τ)v, ℘(v, τ) = −θ
′′
1(v, τ)
θ1(v, τ)
+
(
θ′1(v, τ)
θ1(v, τ)
)2
− 4πig1(τ),
in the identity [22]
[ζ(x) + ζ(y) + ζ(z)]2 = ℘(x) + ℘(y) + ℘(z),
we get(
θ′1(x, τ)
θ1(x, τ)
+
θ′1(y, τ)
θ1(y, τ)
+
θ′1(z, τ)
θ1(z, τ)
)2
= −12πig1(τ)− θ′′1(x, τ)
θ1(x, τ)
+
θ′
2
1 (x, τ)
θ21(x, τ)
− θ′′1(y, τ)
θ1(y, τ)
+
θ′
2
1 (y, τ)
θ21(y, τ)
− θ′′1(z, τ)
θ1(z, τ)
+
θ′
2
1 (z, τ)
θ21(z, τ)
.
Simplifying
2
θ′1(x− y, τ)
θ1(x− y, τ)
[
θ′1(x, τ)
θ1(x, τ)
− θ′1(y, τ)
θ1(y, τ)
]
+ 2
θ′1(x, τ)
θ1(x, τ)
θ′1(y, τ)
θ1(y, τ)
= 3
η
ω
− θ′′1(x, τ)
θ1(x, τ)
− θ′′1(y, τ)
θ1(y, τ)
− θ′′1(z, τ)
θ1(z, τ)
,
using the fact that
4πi
∂τθ
′
1(0, τ)
θ′1(0, τ)
= −12πig1(τ),
∂2
∂2v
θ1(v, τ) = 4πi
∂
∂τ
θ1(v, τ), (A.3)
and doing the substitution y 7→ −y, z 7→ x− y, we get the desired identity. �
Substituting in the lemma x = v2, y = −v2 we get
2
θ′′1(v2, τ)
θ1(v2, τ)
+ 2
θ′
2
1 (v2, τ)
θ21(v2, τ)
= 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ1(2v2, τ)
))
+ 4
θ′1(2v2, τ)
θ1(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
. (A.4)
Substituting (A.4) in (A.1)
g13 = −2πit2
∂
∂τ
(
log
(
θ′1(0, τ)
θ1(2v2, τ)
))
.
Computing g22:
g22 =
1
2
(
∂t2
∂v0
)2
− 1
2
(
∂t2
∂v2
)2
+ 2
∂t2
∂u
∂t2
∂τ
=
1
2
(
∂t2
∂v0
)2
− 1
2
(
∂t2
∂v2
)2
− 4πit2
∂t2
∂τ
.
32 G.F. Almeida
First, we separately compute ∂t2
∂v2
, ∂t2
∂v0
, ∂t2
∂τ
1
2
(
∂t2
∂v0
)2
=
(
t2
)2
2
[
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
+
θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
]2
,
−1
2
(
∂t2
∂v2
)2
= −
(
t2
)2
2
[
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]2
,
−4πit2
∂t2
∂τ
= −4πi
(
t2
)2
2
[
∂τθ1(v0 + v2, τ)
θ1(v0 + v2, τ)
+
∂τθ1(v0 − v2, τ)
θ1(v0 − v2, τ)
− ∂τθ1(2v2, τ)
θ1(2v2, τ)
]
− 4πi
(
t2
)2
2
[
−∂τθ
′
1(0, τ)
θ′1(0, τ)
]
.
Summing the equations we get
g22 =
(
t2
)2
2
[
4
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
]
+
(
t2
)2
2
[
4
θ′1(2v2, τ)
θ1(2v2, τ)
[
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
]
− 4
θ′
2
1 (2v2, τ)
θ21(2v2, τ)
]
+
(
t2
)2
2
[
−2
θ′′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′′1(v0 − v2, τ)
θ1(v0 − v2, τ)
− 8πi
[
−∂τθ1(2v2, τ)
θ1(2v2, τ)
− ∂τθ
′
1(0, τ)
θ′1(0, τ)
]]
,
where was used (A.3). Substituting in Lemma A.1 x = v0 + v2, y = v0 − v2 we get
θ′′1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
= 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(2v2, τ)
))
+ 2
θ′1(2v2, τ)
θ1(2v2, τ)
[
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
]
.
Substituting the last identity in g22 we get
g22 = 2
(
t2
)2 [θ′′1(2v2, τ)
θ1(2v2, τ)
− θ′
2
1 (2v2, τ)
θ21(2v2, τ)
]
.
Computing g12:
g12 =
1
2
∂t1
∂v0
∂t2
∂v0
− 1
2
∂t1
∂v2
∂t2
∂v2
+
∂t1
∂u
∂t2
∂τ
+
∂t2
∂u
∂t1
∂τ
=
1
2
∂t1
∂v0
∂t2
∂v0
− 1
2
∂t1
∂v2
∂t2
∂v2
− 2πit2
∂t1
∂τ
− 2πit1
∂t2
∂τ
.
We have that
∂t1
∂v0
= 2
θ′1(v0, τ)
θ1(v0, τ)
θ21(v0, τ)
θ21(v2, τ)
e−2πiu + 2
∂t2
∂v0
θ′1(v2, τ)
θ1(v2, τ)
,
∂t1
∂v2
= −2
θ′1(v2, τ)
θ1(v2, τ)
θ21(v0, τ)
θ21(v2, τ)
e−2πiu + 2
∂t2
∂v2
θ′1(v2, τ)
θ1(v2, τ)
+ 2t2
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
,
∂t1
∂τ
= 2
[
∂τθ1(v0, τ)
θ1(v0, τ)
− ∂τθ1(v2, τ)
θ1(v2, τ)
]
θ21(v0, τ)
θ21(v2, τ)
e−2πiu+ 2
∂t2
∂τ
θ′1(v2, τ)
θ1(v2, τ)
+ 2t2
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 33
Therefore
1
2
∂t1
∂v0
∂t2
∂v0
= t2
[
θ′1(v0+v2, τ)
θ1(v0+v2, τ)
+
θ′1(v0−v2, τ)
θ1(v0−v2, τ)
]
θ′1(v0, τ)
θ1(v0, τ)
θ21(v0, τ)
θ21(v2, τ)
e−2πiu+
(
∂t2
∂v0
)2
θ′1(v2, τ)
θ1(v2, τ)
,
−1
2
∂t1
∂v2
∂t2
∂v2
= −t2
[
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
]
θ′1(v2, τ)
θ1(v2, τ)
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
− t2
[
−2
θ′1(2v2, τ)
θ1(2v2, τ)
]
θ′1(v2, τ)
θ1(v2, τ)
θ21(v0, τ)
θ21(v2, τ)
e−2πiu −
(
∂t2
∂v2
)2
θ′1(v2, τ)
θ1(v2, τ)
− t2 ∂t
2
∂v2
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
,
−2πit1
∂t2
∂τ
= −2πi
[
∂τθ1(v0 + v2, τ)
θ1(v0 + v2, τ)
+
∂τθ1(v0 − v2, τ)
θ1(v0 − v2, τ)
]
t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
− 2πi
[
−∂τθ1(2v2, τ)
θ1(2v2, τ)
− ∂τθ
′
1(0, τ)
θ′1(0, τ)
]
t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu − 4πit2
∂t2
∂τ
θ′1(v2, τ)
θ1(v2, τ)
,
−2πit2
∂t1
∂τ
= −4πit2
[
∂τθ1(v0, τ)
θ1(v0, τ)
− ∂τθ1(v2, τ)
θ1(v2, τ)
]
θ21(v0, τ)
θ21(v2, τ)
e−2πiu − 4πit2
∂t2
∂τ
θ′1(v2, τ)
θ1(v2, τ)
− 4πi
(
t2
)2 ∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
.
Let us separate g12 in three terms
g12 = (1) + (2) + (3),
where
(1) = t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
θ′1(v0, τ)
θ1(v0, τ)
(
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
+
θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
)]
+ t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
θ′1(v2, τ)
θ1(v2, τ)
(
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
)]
+ t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
−2πi
(
∂τθ1(v0 + v2, τ)
θ1(v0 + v2, τ)
+
∂τθ1(v0 − v2, τ)
θ1(v0 − v2, τ)
)]
+ t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
−2πi
(
−∂τθ1(2v2, τ)
θ1(2v2, τ)
− ∂τθ
′
1(0, τ)
θ′1(0, τ)
)]
+ t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
−4πi
(
∂τθ1(v0, τ)
θ1(v0, τ)
− ∂τθ1(v2, τ)
θ1(v2, τ)
)]
,
(2) =
θ′1(v2, τ)
θ1(v2, τ)
[(
∂t2
∂v0
)2
−
(
∂t2
∂v2
)2
− 8πit2
∂t2
∂τ
]
= 4
θ′1(v2, τ)
θ1(v2, τ)
(
t2
)2 [θ′′1(2v2, τ)
θ1(2v2, τ)
− θ′
2
1 (2v2, τ)
θ21(2v2, τ)
]
,
where was used the previous computation of g22
(3) = −4πi
(
t2
)2 ∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
− t2 ∂t
2
∂v2
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
.
To simplify the expression (1) we need to use the Lemma A.1 with the following substitutions
x = v0, y = v2
θ′′1(v0, τ)
θ1(v0, τ)
+
θ′′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(v0, τ)
θ1(v0, τ)
θ′1(v2, τ)
θ1(v2, τ)
34 G.F. Almeida
= 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(v0 − v2, τ)
))
+ 2
θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
[
θ′1(v0, τ)
θ1(v0, τ)
− θ′1(v2, τ)
θ1(v2, τ)
]
. (A.5)
Using the substitutions x = v0, y = −v2
θ′′1(v0, τ)
θ1(v0, τ)
+
θ′′1(v2, τ)
θ1(v2, τ)
+ 2
θ′1(v0, τ)
θ1(v0, τ)
θ′1(v2, τ)
θ1(v2, τ)
= 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(v0 + v2, τ)
))
+ 2
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
[
θ′1(v0, τ)
θ1(v0, τ)
+
θ′1(v2, τ)
θ1(v2, τ)
]
. (A.6)
Summing (A.5) with (A.6)
2
θ′′1(v0, τ)
θ1(v0, τ)
+ 2
θ′′1(v2, τ)
θ1(v2, τ)
− 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(v0 − v2, τ)
))
− 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(v0 + v2, τ)
))
= 2
θ′1(v0, τ)
θ1(v0, τ)
(
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
+
θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
)
+ 2
θ′1(v2, τ)
θ1(v2, τ)
(
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
)
.
Substituting in (1) we get
(1) = t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
−2
θ′1(2v2, τ)
θ1(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
]
+ t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
−2πi
(
−∂τθ1(2v2, τ)
θ1(2v2, τ)
+
∂τθ
′
1(0, τ)
θ′1(0, τ)
)
+ 8πi
∂τθ1(v2, τ)
θ1(v2, τ)
]
= t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
−2
θ′1(2v2, τ)
θ1(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
− 2πi
∂
∂τ
(
log
θ′1(0, τ)
θ1(2v2, τ)
)
+ 2
θ′′1(v2, τ)
θ1(v2, τ)
]
.
Using the identity (A.2), we get
(1) = t2
θ21(v0, τ)
θ21(v2, τ)
e−2πiu
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
.
We compute (3)
(3) = −4πi
(
t2
)2 ∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
− t2 ∂t
2
∂v2
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
= −4πi
(
t2
)2 ∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
− t2
(
t1 − 2t2
θ′1(2v2, τ)
θ1(2v2, τ)
)[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
= −4πi
(
t2
)2 ∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
+ 2
(
t2
)2 θ′1(2v2, τ)
θ1(2v2, τ)
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
− t2 θ
2
1(v0, τ)
θ21(v2, τ)
e−2πiu
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ
′2
1 (v2, τ)
θ21(v2, τ)
]
−2
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ
′2
1 (v2, τ)
θ21(v2, τ)
]
.
The result implies
(1) + (3) = −4πi
(
t2
)2 ∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
− 2
(
t2
)2 [θ′1(v2, τ)
θ1(v2, τ)
− θ′1(2v2, τ)
θ1(2v2, τ)
][
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 35
Computing g12:
g12 = −4πi
(
t2
)2 ∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
− 2
(
t2
)2 [θ′1(v2, τ)
θ1(v2, τ)
− θ′1(2v2, τ)
θ1(2v2, τ)
][
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
+ 4
θ′1(v2, τ)
θ1(v2, τ)
(
t2
)2 [θ′′1(2v2, τ)
θ1(2v2, τ)
− θ′
2
1 (2v2, τ)
θ21(2v2, τ)
]
.
To simplify this expression we need to prove one more lemma.
Lemma A.2.
2
θ′′′1 (v2, τ)
θ1(v2, τ)
+ 2
θ′′1(v2, τ)θ′1(v2, τ)
θ1(v2, τ)
− 4
θ′
3
1 (v2, τ)
θ31(v2, τ)
= 4πi
∂2
∂v2∂τ
(
log
(
θ′1(0, τ)
θ1(2v2, τ)
))
+ 8
θ′′1(2v2, τ)
θ1(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
− 8
θ′
2
1 (2v2, τ)
θ21(2v2, τ)
θ′1(v2, τ)
θ1(v2, τ)
+ 4
θ′1(2v2, τ)
θ1(2v2, τ)
θ′′1(v2, τ)
θ1(v2, τ)
− 4
θ′1(2v2, τ)
θ1(2v2, τ)
θ′
2
1 (v2, τ)
θ21(v2, τ)
. (A.7)
Proof. Differentiating the identity with respect to v2 we obtain (A.7). �
Computing g12:
g12 =
(
t2
)2 [−θ′′′1 (v2, τ)
θ1(v2, τ)
+
θ′1(v2, τ)θ′′1(v2, τ)
θ21(v2, τ)
− 2
θ′1(v2, τ)
θ1(v2, τ)
θ′′1(v2, τ)
θ1(v2, τ)
+ 2
θ′
3
1 (v2, τ)
θ31(v2, τ)
]
+
(
t2
)2 [
2
θ′1(2v2, τ)
θ1(2v2, τ)
θ′′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
θ′
2
1 (v2, τ)
θ21(v2, τ)
+ 4
θ′1(v2, τ)
θ1(v2, τ)
θ′′1(2v2, τ)
θ1(2v2, τ)
]
+
(
t2
)2 [−4
θ′1(v2, τ)
θ1(v2, τ)
θ′
2
1 (2v2, τ)
θ21(2v2, τ)
]
.
Applying (A.7), we get
g12 = −2πi
(
t2
)2 [ ∂2
∂v2∂τ
(
log
(
θ′1(0, τ)
θ1(2v2, τ)
))]
.
Computing g11:
g11 =
1
2
(
∂t1
∂v0
)2
− 1
2
(
∂t1
∂v2
)2
+ 2
∂t1
∂u
∂t1
∂τ
=
1
2
(
∂t1
∂v0
)2
− 1
2
(
∂t1
∂v2
)2
− 4πit1
∂t1
∂τ
.
Computing 1
2
(
∂t1
∂v0
)2
, 1
2
(
∂t1
∂v2
)2
and −4πit1 ∂t
1
∂τ :
To simplify the computation let us define
A :=
θ21(v0, τ)
θ21(v2, τ)
e−2πiu.
Then,
1
2
(
∂t1
∂v0
)2
= 2
θ′
2
1 (v0, τ)
θ21(v0, τ)
A2 + 4A
θ′1(v0, τ)
θ1(v0, τ)
∂t2
∂v0
θ′1(v2, τ)
θ1(v2, τ)
+ 2
(
∂t2
∂v0
)2
θ′
2
1 (v2, τ)
θ21(v2, τ)
,
−1
2
(
∂t1
∂v2
)2
= −2
θ′
2
1 (v2τ)
θ21(v2, τ)
A2 + 2A
θ′1(v2, τ)
θ1(v2, τ)
[
2
∂t2
∂v2
θ′1(v2, τ)
θ1(v2, τ)
+ 2t2
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]]
36 G.F. Almeida
− 2
(
∂t2
∂v2
)2
θ′
2
1 (v2, τ)
θ21(v2, τ)
− 4t2
∂t2
∂v2
θ′1(v2, τ)
θ1(v2, τ)
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
− 2
(
t2
)2 [θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]2
,
−4πit1
∂t1
∂τ
= −8πiA2
[
∂τθ1(v0, τ)
θ1(v0, τ)
− ∂τθ1(v2, τ)
θ1(v2, τ)
]
− 8πiA
∂t2
∂τ
θ′1(v2, τ)
θ1(v2, τ)
− 8πiAt2
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
− 16πiAt2
θ′1(v2, τ)
θ1(v2, τ)
[
∂τθ1(v0, τ)
θ1(v0, τ)
− ∂τθ1(v2, τ)
θ1(v2, τ)
]
− 16πit2
∂t2
∂τ
θ′
2
1 (v2, τ)
θ21(v2, τ)
− 16πi
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
.
Then, we have
g11 = (1) + (2) + (3) + (4) + (5),
where
(1) = A2
[
2
θ′
2
1 (v0, τ)
θ21(v0, τ)
− 2
θ′
2
1 (v2τ)
θ21(v2, τ)
− 8πi
[
∂τθ1(v0, τ)
θ1(v0, τ)
− ∂τθ1(v2, τ)
θ1(v2, τ)
]]
= A2
[
2
θ′
2
1 (v0, τ)
θ21(v0, τ)
− 2
θ′
2
1 (v2τ)
θ21(v2, τ)
− 2
θ′′1(v0, τ)
θ1(v0, τ)
+ 2
θ′′1(v2, τ)
θ1(v2, τ)
]
= 2A2[℘(v0)− ℘(v2)] = 2
16ω4
[℘(v0)− ℘(v2)]2
[℘(v0)− ℘(v2)] = 32
ω4
℘(v0)− ℘(v2)
,
(2) = −8πit2A
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
+ 2At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
2
θ′1(v0, τ)
θ1(v0, τ)
[
θ′1(v0−v2, τ)
θ1(v0−v2, τ)
+
θ′1(v0+v2, τ)
θ1(v0+v2, τ)
]]
+ 2At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
2
θ′1(v2, τ)
θ1(v2, τ)
[
−θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]]
+ 2At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
2
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
− 8πi
[
∂τθ1(v0, τ)
θ1(v0, τ)
− ∂τθ1(v2, τ)
θ1(v2, τ)
]]
+2At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
−4πi
[
∂τθ1(v0+v2, τ)
θ1(v0+v2, τ)
+
∂τθ1(v0−v2, τ)
θ1(v0−v2, τ)
− ∂τθ1(2v2, τ)
θ1(2v2, τ)
− ∂τθ
′
1(0, τ)
θ′1(0, τ)
]]
.
Using (A.2),
2
θ′′1(v0, τ)
θ1(v0, τ)
+ 2
θ′′1(v2, τ)
θ1(v2, τ)
− 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(v0 − v2, τ)
))
− 4πi
∂
∂τ
(
log
(
θ′1(0, τ)
θ(v0 + v2, τ)
))
= 2
θ′1(v0, τ)
θ1(v0, τ)
(
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
+
θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
)
+ 2
θ′1(v2, τ)
θ1(v2, τ)
(
−θ
′
1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
)
,
(2) = −8πit2A
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
+ 2At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
−4
θ′1(v2, τ)
θ1(v2, τ)
θ′1(2v2, τ)
θ1(2v2, τ)
]
+ 2At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
2
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
+ 4
θ′′1(v2, τ)
θ1(v2, τ)
]
+ 2At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
−4πi
∂τθ
′
1(0, τ)
θ′1(0, τ)
+ 4πi
∂τθ1(2v2, τ)
θ1(2v2, τ)
]
.
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 37
Using again (A.2),
(2) = −8πit2A
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
+ 8At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
,
(3) = 4
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
1
2
(
∂t2
∂v0
)2
− 1
2
(
∂t2
∂v2
)2
− 4πit2
∂t2
∂τ
]
= 8
θ′
2
1 (v2, τ)
θ21(v2, τ)
(
t2
)2 [θ′′1(2v2, τ)
θ1(2v2, τ)
− θ′
2
1 (2v2, τ)
θ21(2v2, τ)
]
,
(4) = −2
(
t2
)2 [ ∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)]2
− 16πi
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
,
(5) = −4
(
t2
) ∂t2
∂v2
θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)
= −4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
−θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
= −4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
−θ′1(v0 − v2, τ)
θ1(v0 − v2, τ)
+
θ′1(v0 + v2, τ)
θ1(v0 + v2, τ)
− 2
θ′1(v2, τ)
θ1(v2, τ)
]
− 4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
= −4
(
t2
)
A
θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)
− 4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
.
Summing (2) and (5)
(2) + (5) = −8πit2A
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
+ 8At2
θ′
2
1 (v2, τ)
θ21(v2, τ)
[
θ′′1(v2, τ)
θ1(v2, τ)
− θ′
2
1 (v2, τ)
θ21(v2, τ)
]
− 4
(
t2
)
A
θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)
− 4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
= −4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
+At2
[
− 2
θ′′′1 (v2, τ)
θ1(v2, τ)
+ 6
θ′1(v2, τ)θ′′1(v2, τ)
θ1(v2, τ)
− 4
θ′
3
1 (v2, τ)
θ31(v2, τ)
]
= −4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
+ 2At2℘′(v2)
= −4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
− 32
ω4
℘(v0)−℘(v2)
.
Summing (1) and (2) + (5)
(1) + (2) + (5) = −4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
.
From the above results, we find
g11 =(1) + (2) + (5) + (3) + (4)
38 G.F. Almeida
=− 4
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)[
2
θ′1(v2, τ)
θ1(v2, τ)
− 2
θ′1(2v2, τ)
θ1(2v2, τ)
]
+ 8
θ′
2
1 (v2, τ)
θ21(v2, τ)
(
t2
)2 [θ′′1(2v2, τ)
θ1(2v2, τ)
− θ′
2
1 (2v2, τ)
θ21(2v2, τ)
]
− 2
(
t2
)2 [ ∂
∂v2
(
θ′1(v2, τ)
θ1(v2, τ)
)]2
− 16πi
(
t2
)2 θ′1(v2, τ)
θ1(v2, τ)
∂
∂τ
(
θ′1(v2, τ)
θ1(v2, τ)
)
.
Summarizing, we have proved the identities (3.7) and (3.8).
Acknowledgements
I am grateful to Professor Boris Dubrovin for proposing this problem, for his remarkable advi-
ses and guidance. I would like also to thanks Professors Davide Guzzetti and Marco Bertola
for helpful discussions, and guidance of this paper. In addition, I thank the anonymous referees
for their valuable comments and remarks, that have helped to improve the paper.
References
[1] Almeida G.F., Differential geometry of orbit space of extended affine Jacobi group An, arXiv:2004.01780.
[2] Bertola M., Frobenius manifold structure on orbit space of Jacobi groups. I, Differential Geom. Appl. 13
(2000), 19–41.
[3] Bertola M., Frobenius manifold structure on orbit space of Jacobi groups. II, Differential Geom. Appl. 13
(2000), 213–233.
[4] Bourbaki N., Lie groups and Lie algebras, Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag,
Berlin, 2002.
[5] Cutimanco M., Shramchenko V., Explicit examples of Hurwitz Frobenius manifolds in genus one, J. Math.
Phys. 61 (2020), 013501, 20 pages.
[6] Dubrovin B., Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups (Mon-
tecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120–348, arXiv:hep-
th/9407018.
[7] Dubrovin B., Differential geometry of the space of orbits of a Coxeter group, in Surveys in Differential Ge-
ometry: Integrable Systems, Surv. Differ. Geom., Vol. 4, Int. Press, Boston, MA, 1998, 181–211, arXiv:hep-
th/9303152.
[8] Dubrovin B., Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of
solutions, in New Trends in Mathematical Physics, Springer, Dordrecht, 2009, 231–276.
[9] Dubrovin B., Strachan I.A.B., Zhang Y., Zuo D., Extended affine Weyl groups of BCD-type: their Frobenius
manifolds and Landau–Ginzburg superpotentials, Adv. Math. 351 (2019), 897–946, arXiv:1510.08690.
[10] Dubrovin B., Zhang Y., Extended affine Weyl groups and Frobenius manifolds, Compositio Math. 111
(1998), 167–219, arXiv:hep-th/9611200.
[11] Dubrovin B., Zhang Y., Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–
Witten invariants, arXiv:math.DG/0108160.
[12] Eichler M., Zagier D., The theory of Jacobi forms, Progress in Mathematics, Vol. 55, Birkhäuser Boston,
Inc., Boston, MA, 1985.
[13] Ferapontov E.V., Pavlov M.V., Xue L., Second-order integrable Lagrangians and WDVV equations,
arXiv:2007.03768.
[14] Hertling C., Multiplication on the tangent bundle, arXiv:math.AG/9910116.
[15] Pavlov M.V., Tsarev S.P., Tri-Hamiltonian structures of Egorov systems of hydrodynamic type, Funct. Anal.
Appl. 37 (2003), 32–45.
[16] Romano S., 4-dimensional Frobenius manifolds and Painleve’ VI, Math. Ann. 360 (2014), 715–751,
arXiv:1209.3959.
https://arxiv.org/abs/2004.01780
https://doi.org/10.1016/S0926-2245(00)00026-7
https://doi.org/10.1016/S0926-2245(00)00027-9
https://doi.org/10.1063/1.5117782
https://doi.org/10.1063/1.5117782
https://doi.org/10.1007/BFb0094793
https://arxiv.org/abs/hep-th/9407018
https://arxiv.org/abs/hep-th/9407018
https://doi.org/10.4310/SDG.1998.v4.n1.a4
https://arxiv.org/abs/hep-th/9303152
https://arxiv.org/abs/hep-th/9303152
https://doi.org/10.1007/978-90-481-2810-5_18
https://doi.org/10.1016/j.aim.2019.05.030
https://arxiv.org/abs/1510.08690
https://doi.org/10.1023/A:1000258122329
https://arxiv.org/abs/hep-th/9611200
https://arxiv.org/abs/math.DG/0108160
https://doi.org/10.1007/978-1-4684-9162-3
https://doi.org/10.1007/978-1-4684-9162-3
https://arxiv.org/abs/2007.03768
https://arxiv.org/abs/math.AG/9910116
https://doi.org/10.1023/A:1022971910438
https://doi.org/10.1023/A:1022971910438
https://doi.org/10.1007/s00208-013-0987-1
https://arxiv.org/abs/1209.3959
The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1 39
[17] Romano S., Frobenius structures on double Hurwitz spaces, Int. Math. Res. Not. 2015 (2015), 538–577,
arXiv:1210.2312.
[18] Saito K., Yano T., Sekiguchi J., On a certain generator system of the ring of invariants of a finite reflection
group, Comm. Algebra 8 (1980), 373–408.
[19] Shramchenko V., Deformations of Frobenius structures on Hurwitz spaces, Int. Math. Res. Not. 2005 (2005),
339–387, arXiv:math-ph/0408026.
[20] Strachan I.A.B., Frobenius submanifolds, J. Geom. Phys. 38 (2001), 285–307, arXiv:math.DG/9912081.
[21] Strachan I.A.B., Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures, Dif-
ferential Geom. Appl. 20 (2004), 67–99, arXiv:math.DG/0201039.
[22] Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge
University Press, Cambridge, 1996.
[23] Wirthmüller K., Root systems and Jacobi forms, Compositio Math. 82 (1992), 293–354.
[24] Zuo D., Frobenius manifolds and a new class of extended affine Weyl groups of A-type, Lett. Math. Phys.
110 (2020), 1903–1940, arXiv:1905.09470.
https://doi.org/10.1093/imrn/rnt215
https://arxiv.org/abs/1210.2312
https://doi.org/10.1080/00927878008822464
https://doi.org/10.1155/IMRN.2005.339
https://arxiv.org/abs/math-ph/0408026
https://doi.org/10.1016/S0393-0440(00)00064-4
https://arxiv.org/abs/math.DG/9912081
https://doi.org/10.1016/j.difgeo.2003.10.001
https://doi.org/10.1016/j.difgeo.2003.10.001
https://arxiv.org/abs/math.DG/0201039
https://doi.org/10.1017/CBO9780511608759
https://doi.org/10.1017/CBO9780511608759
https://doi.org/10.1007/s11005-020-01280-2
https://arxiv.org/abs/1905.09470
1 Introduction
1.1 Orbit space of reflection groups and its extensions
1.2 Hurwtiz space/orbit space correspondence
1.3 Results
2 Invariant theory of J(A1)
2.1 The group J(A1)
2.2 Jacobi forms of J(A1)
3 Frobenius structure on the orbit space of Ja(A1)
3.1 Intersection form
3.2 Euler and unit vector field
3.3 Flat coordinates of the Saito metric
3.4 Construction of WDVV solution
4 Conclusion
A Appendix
References
|
| id | nasplib_isofts_kiev_ua-123456789-211166 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T18:30:06Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Almeida, Guilherme F. 2025-12-25T13:20:16Z 2021 The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁. Guilherme F. Almeida. SIGMA 17 (2021), 022, 39 pages 1815-0659 2020 Mathematics Subject Classification: 53D45 arXiv:1907.01436 https://nasplib.isofts.kiev.ua/handle/123456789/211166 https://doi.org/10.3842/SIGMA.2021.022 We define certain extensions of Jacobi groups of A₁, prove an analogue of the Chevalley theorem for their invariants, and construct a Dubrovin-Frobenius structure on their orbit space. I am grateful to Professor Boris Dubrovin for proposing this problem, for his remarkable advice and guidance. I would also like to thank Professors Davide Guzzetti and Marco Bertola for helpful discussions and guidance on this paper. In addition, I thank the anonymous referees for their valuable comments and remarks, which have helped to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁ Article published earlier |
| spellingShingle | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁ Almeida, Guilherme F. |
| title | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁ |
| title_full | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁ |
| title_fullStr | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁ |
| title_full_unstemmed | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁ |
| title_short | The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A₁ |
| title_sort | differential geometry of the orbit space of extended affine jacobi group a₁ |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211166 |
| work_keys_str_mv | AT almeidaguilhermef thedifferentialgeometryoftheorbitspaceofextendedaffinejacobigroupa1 AT almeidaguilhermef differentialgeometryoftheorbitspaceofextendedaffinejacobigroupa1 |